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-----­Physics for Everyone

Book 4

AI I, Kitaigorodsky

PHOTONS ANDNUCLE

Translated from

the Russian

by George Yankovsky

Mir Publishc~rsMoscow

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PREFACE

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First published 1981

© Ib!13Tf',clhCTBO «flayKa,'. 1'17<)© English translation,

Mir Publishers, 1981

This is the fourth and concluding book in the Physicsfor Everyone series and it deals with the fundamentalsof physics.

"Fundamentals" is of course a rather vague word butwe will think of it as meaning the general laws on whichthe whole edifice of modern physics rests. There are notso many of them, and so we can make a list: the laws ofmotion of classical mechanics, the laws of thermody­namics, the laws that are embodied in the equations ofMaxwell and that govern charges, currents and electro­magnetic fields, and then the laws of quantum physicsand the theory of relativity.

The laws of physics, like those of natural science atlarge, are of an empirical nature. They are arrived atby means of observation and experiment. Experimentsestablish a multitude of primary facts such as the build­ing lip of matter from atoms and molecules, the nuclearmodel of the atom, the wave-particle aspect of matter,and so on. Now, both the number of basic laws and alsothe number of fundamental facts and concepts necessaryfor their description is not so very great: At any rate,it is limited.

During the past several decades, physics has grownand expanded to such an extent that workers in differentbl'anches cease to understand one another as soon as the(.1 isc~lssion goes beyond what holds them together in onelamdy, that is, beyond the limits of the laws and COIl­cppls. underlying all branches of pllysics. Portions ofphYSICS are closely interwoven with technology, with otherareas of natnral science, with medicine, and even withjill' humanitarian sciences. It is easy to sec WilY theyha\:u set tltumsclves lip as indqwndent disciplines.

:::;~Il'uly no one wOllld argue tllat allY discussion of thevanous divisions of applied physics must be preceded

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A. I. Kitaigorodsky

bv an examination of the basic laws and facts. And it isj~st as true that different writers select and arrange thematerial needed for laying the foundation of physicseach in his own way, depending on his individual tastesand his own special field of inquiry. What I have tooffer here is merely one of many possible expositionsof the fundamentals of physics.

The type of reader envisaged by this Physics tor E very­one series has been mentioned in the prefaces to the ear­lier books. I will repeat that this series is aimed at repre­sentatives of all professions who wish to recall the phys­ics they studied, get a picture of the modern state ofthe science, aud evaluate the effect it has on scientificand technological progress and on forming a materialistworld outlook. Many pages of these books will, I amsure, be of interest to teachers of physics and to studentsat school that have come to like physics. And finallythere may be something of interest for those readerswho are depressed by even a simple algebraic equation.

Quite naturally, this series is not intended to takethe place of a textbook. Photons and Nu':lei is an attempt,on the part of the author, to demonstrate to the readel'how the laws of the electromagnetic field and quantumphysics operate when we consider the behaviour of elec­tromagnetic waves of different wavelength. Before takingup atomic nuclei, the reader will get some idea of whatwave mechanics and the special theory of relativity aroabout. This is followed bv a discussion of the basic factsconcerning the strueturo 'of atomic nuclei, and thon tltotopic will he sources of energy on tho eal'tll-a topic ofburning interest to humanity at large. vVe conclude ourbrief talk with a story about the physics of the universe.

The lim ited scope of this book has forced us to giveup lllauy tmditiollal topics. '1'110 old 1l111St always giveway to the new.

Preface 6

CONTENTS

Preface

1. Soft Electromagnetic RadiationExchange of Energy by Radiation 9. The Radiationof Incandescent Bodies 11. The Theory of ThermalRadiation 17. Optical Spectra 20. Laser Radiation 27.Luminescence 37.

2. Optical InstrumentsThe Prism 40. The Lens 44. The Camera 48. The Eye 51.Polarizers 53. The Microscope and the Telescope 56.Interferometers 61. Laser Devices 12. Photometry 74.Holography 78.

3. Hard Electromagnetic RadiationThe Discovery of X Rays 82. X-ray Diffraction Analysis88. The X-ray Spectrum 98. Radiography of Materials103.

4. Generalizations of MechanicsRelativistic Mechanics 108. Particles with VelocitiesClose to the Velocity of Light 121. Wave Mechanics 127.

The Heisenberg Uncertainty Principle 131.

S. The Structure of Atomic NucleiIsotopes 137. Radioactivity 142. Radioactive Decay 147.Nuclear Reactions and the Discovery of the Neutron 151.

Contents

Properties of Atomic Nuclei 153. Bosons and Fermlons 156.The Mass and Energy of an Atomic Nucleus 160.The Energy of Nuclear Reactions 163. A Nuclear ChainReaction 166.

6. Energy Around UsSources of Energy 172. Fuel 178. Eleefric PowerPlants 182. Nuclear Reactors 188. Thermonuclear Energy197. Solar Energy 201. Power from the Wind 206.

7. The Physics of the UniverseMeasuring Distance to the Stars 219. The ExpandingUniverse 215. The General Theory of Relativity 220.Stars of All Ages 225. Radio Astronomy 230. CosmicRays 233.

8l. ,Soft ElectromagneticRadiation

Exchange of Energy by Radiafion

Soft electromagnetic radiation is that with wavelengthslying roughly in the interval from 0.1 to 100 micrometres.Also, ~ear in mind that when we speak of soft radiationwe mean electromagnetic waves not dealt with in radioengineering. This stipulation is important because purelyradio-engineering methods permit one to dip into theregion of soft radiation. The soft radiation also ratherfrequently goes by the simple term "light". When applyingthat term, one must bear in mind that visible light occupiesonly avery narrow section of wavelengths-for the "ave­rage" human eye it lies between 380 and 780 nanometres(or from 0.38 to 0.78 micrometres).

In future, whenever we have the occasion to speak of"light", we will do so in the broad sense of the wordbecause the laws that hold true for the visible portion ofthe spectrum remain true for all other representatives ofsoft radiation.

Also note that radiation with shorter wavelenoths thanvisible light is called ultra vio le t radiation; tl~e longerwavelengths are termed infrared radiation .

.We can now turn to the topic of our discussion. It~hll be recalled that there are three modes of heat transfer.

ey are called heat conduction, thermal convectionand thermal radiation. In order to study the exchange of:netgr that occurs in thermal radiation, we will have tolCamlue the behaviour of ,hodies in a vac uum (where con-

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Photons and Nuclei 10

vection is impossible) separated by a certain distance(this is to exclude the conduction of heat).

Experiments have shown that if two or more bodiesform a closed system (the reader will recall that thismeans the absence of any exchange of energy betweenobjects not in the system), the temperatures of the bodiesequal out. Each one of the bodies of the system is atthe same time a radiator and an absorber. What occurare numberless transitions of atoms and molecules froma higher level to a lower level (such events involve theemission of photons) and from ~ lower level to a higherlevel (photons are absorbed). Photons of all energies (or,what is the same thing, electromagnetic waves of allwavelengths) participate in these exchanges.

Quite naturally, the body does not absorb all the energyfalling on it. There mlly be bodies that scatter moreenergy or t.ransmit certain wavelengths. But this is of noconsequence because sooner or later a thermal equilibriumis established nevertheless.

The condition of thermal equilibrium requires thatthe ratio of the energy of absorption to the energy ofemission be the same for all wavelepgths. This theoremwas rigorously demonstrated in 1860 by the German phys­icist Gustav Robert Kirchhoff (1824-1887). The ratio canchange for different temperatures, but if the temperatureis fixed, then it is the same for photons of all energies.

This is a clear enough theorem and hardly needs anydemonstration of proof. The idea behind the law is thatthe number of absorbed photons of a given kind (that is,of a definite energy) is equal, in the case of thermal equi­librium, to the number of radiated photons of that par­ticular .kind. From this we get the following rule: if anobject is a strong absorber of any kind of rays, thenthose same rays are just as strongly radiated.

This rule helps to predict the conditions under whichthermal equilibrium sets in. Why is water in a bottle

t. Soft Eledromagnetic Radiation 11

with silvered sides so slow to heat up under the actionof the sun's rays, whereas water in a flask made of blackglass heats up very quickly? The explanation is obvious:a black body absorbs rays intensively and their energygoes to increase the temperature, and thermal equilib­rium sets in after intense heating. A silvered surface, on thecontrary, is an excellent reflector. Only a small amountof the energy is absorbed, it takes a long time to heatthe body, and equilibrium sets in at a low temperature.

Now let's reverse the experiment. Pour some hot waterinto both flasks and put them into a refrigerator. Whichone will cool off quickest? The one that heats up fasterwill cQol off faster. If more energy is absorbed, moreis released.

Some very effective experiments can be performed withcoloured ceramics. If the object is green, the piece absorbsall colours except green. This is because the eye seesthose rays that are reflected (or scattered) by the material.Now heat up the fragment. How will it appear? Theanswer is right at the tip of your tongue: violet becauseviolet is the complementary colour of yellow-green. Com­plementary colours are those that produce white if theyare mixed. Newton was the one who introduced the term"complementary colour" when he decomposed light raysinto a spectrum with the aid of a glass prism.

The Radiation of Incandescent Bodies

It is well known that a piece of metal, when heated,first becomes red and then white. Most chemical substancescannot be thus heated. They either melt or decompose.Therefore, what follows refers mostly to metals.

The most remarkable thing is that the radiation spec­trum of all heated bodies is not at all specific. The pointis this. From the basic law about energy levels it isclear that the radiation spectrum and the absorption

spectrum of a body must coincide. Metals are opaquethroughout the region of the spectrum of soft radiation.From this it follows that they must also radiate photonsof all energies.

Let's put this differently: a continuous spectrum ap­pears due to the fact that in a multi-atomic system theenergy levels of the atoms merge into overlapping bands.In such a system, all energy transitions are possible,that is, we can find any energy difference between themth and nth levels Em-En and, hence, any frequenciesof radiation and absorption. Figure 1.1 shows the spec­trum of an incandescent body for several temperatures(we give here the theoretical curves that hold true fora so-called ideal black body).

It is well to point out here that the derivation of theshape of this curve (this was done by Max Planck in 1900)was the first step in the development of quantum physics.In order to obtain agreement of theory with experiment,Planck had to assume that radiation and absorption oflight take place in separate portions. Planck was notprepared to take the next step and state that we arejustified in speaking of particles of light (photons). Thatstep was taken by Albert Einstein in 1905.

It was only in 1913 that Niels Bohr introduced theconcept of quantization of energy. And if we want alogically rigorous theory of thermal radiation, the yearof birth can be put as 1926.

Let's first discuss the shapes of the curves and onlythen talk about theory. First of all, note that as thetemperature rises, the area under the curve rapidly in­creases. What is the physical meaning of the area boundedby the radiation curve? When constructing curves sim­ilar to those depicted in the figure, we say that theintensity of radiation for a given wavelength is laidoff on the axis of ordinates (horizontal axis). But what,does;a "given wavelength" mean? Do we mean 453 -or

Figure 1.1

453.2 nanometres? Maybe 453.2.57859987654? It is prob­ably cleal' that when we speak of a "given wa"elength",We mean avery small interval 2J wavelengths. We agree,say, that the interval is equa:! to O.Ot nanometre. F~om

this it follows that it is not the ordinate that has phySIcalmeaning but a tiny column with a base of 0.01 nanom.etre.The area of this column is equal to the energy radIatedby the waves having leJ;lgths in that interval (for example,from 453.25 to 453.26 nanometres). Now if we break upthe whole area into such columns, we get the total inten­sity of the whole spectrum. That is precisely the operationmathematicians perform and it is called integration. Tosummarize: the area under our curve yields the total

13

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1. Soft Electromagnetic Radiation12Photons and Nuclei

t. Soft Electromagnetic Radiation 15

intensity of the radiation, and it turns out to be pro­pOPtional to the fourth power of the temperature.

In the figure we are discussing it is clear that with in­'Creasing temperature there is not only a change in thearea oecupied by the curve but there is a shift of its max­imum to the left, that is, into the region of ultravioletradiation.

The relationship between the wavelength of light inmicrometres that corrl.'sponds to the greatest intensityof radiation (or absorption) and the temperature in kel­vins is given by the following formula:

2886A.max ='1'

At the lowest temperatures, the maximum lies in theinfrared region. That is precisely why infrared radiationis also termed thermal radiation. And it is a marvelousthing that we have instruments capable of sensing thethermal radiation emitted by bodies at room temperatureand even lower. There are instruments today that can seein total darkness. Some animals, by the way, have thesame capability. There is nothing at all strange in thisfact since infrared rays have, in principle, the sameproperties as visible rays.

Also, don't forget that every animal is a source ofradiation. We sometimes hear of a person being able to"feel" the presence of another person in darkness. Nomysticism is involved, merely the one who "feels" hasa highly sensitive perception of thermal rays.

, I can't resist telling the reader about an interestingepisode that demonstrates the necessity of taking thermalrays into account even when, in the very ordinary senseof this word, the source of rays is not a heated body. Afew years ago I was asked to investigate some experimentsConducted by a person whl) considered himself a magiciancapable of stopping a motor using only his will power.

14Photons and Nuclei

Max Planck l1858.1947J-outstanding German scientist who laidthe foundations of quantum theory. In an attempt to find a mathe­matical expression for a proper description of the spectral distri­bution of the emission of an ideal black body Planck demonstratedthat such a formula could be obtained by introducing a "quantumof action". Panek assumed that a body emits energy in parcels,equal to the product of a constant (which later was named afterhim) by the frequency of the ligh t.

Figure 1.2

My task was to find a rational expla.nati.o~ (sorc~rers ofthe twentieth like to deal in pseudoscI~ntI~C termmologyand so call thesa experiments telekInesI~). ,

A diagram of the experiment is shown m Figure .1.2.A wing was set in rotation by a small ?1?tor, and thewing really did stop whenever the magician sat downnext to the box where the axle of the motor emergedfrom below, I soon found that anybody ~ho sat downin that position would be able to stop the wmg. It usu~llytook about 10 to 15 minutes to perform the operatIOn.And it wasn't the motor that came to a halt, as the ma-.' claI'med but the little wing. It was clear thengiclan, . h h b d

that some kind of force connected with t e uman 0 ywas interfering with the force of adhesion between theaxle of the motor and the wing.

I pointed out that the wing could be stopped almostinstantly if an electric lamp were brought up close ~o theside of the box. It was' obvious that the tn~k lay m theheat emitted by the human body. I sent a whiff of toba~cosmoke into the box and demonstrated that the convectIOncurrents of air inside the box move so as to prevent thewing from rotating. Exact measurements showed thatthe temperature of the side of the box closest to the human

1. Soft Electromagnetic Radiation 17

body is about one degree higher than the opposite side.Infrared rays emitted by a body heated to 60-70 de­

grees Celsius can be felt if the hand is held at a short dis­tance. Thermal convection of course must be eliminated.!leated air rises upwards, so bring your hand close frombelow. You will then be certain that you are perceivingthermal rays.

We conclude our talk about thermal rays with anexplanation of why the modern electric light bulb witha tungsten filament is a great step beyond a bulb witha carbon filament. The whole point is that a carbonfilament can be heated to 2100 K, while a tungsten fila­ment can be heated all the way up to 2500 K. Why arethese 400 degrees so important? Because the purpose ofan incandescent lamp is to provide light and not heat,and so the aim is to have the maximum of the curvelocated in the visible portion of the radiation. It willbe seen from the graph that the ideal is to have a filamentthat can withstand the temperature of the sun's surface,or 6000 K. But even the step from 2100 degrees to 2500degrees raises the portion of energy involving visibleradiation from 0.5 % to 1.6 %.

The Theory of Thermal Radiation

If a system of radiating and absorbing bodies is closed,then the photon "gas" (with the aid of which the bodie,sexchange energy) must be in equilibrium with the atomsSUpplying the photons. The number of photons withenergy hv depends on how many atoms lie in the E1level and how many in the E 2 level. In the case of equi­librium, these numbers remain unchanged.

However, the equilibrium is of a dynamic nature sincethe processes of excitation and radiation occur at thesame time. In some way (either by collision with anotherParticle or due to absorption of a photon from without)2-2542

16

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Photons and Nuclei

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Photons and Nuclei 18

the atom or the atomic system climbs to a high level.The system persi~ts in this excited state for some (in­definite) time (ordinarily for a fraction of a second) andthen reverts to a low level. This process is termed sponta­neous radiation. The atom behaves like a little ball onthe top of a sharp peak with an intricate configuration:the slightest breath of air is enough to disrupt the equi­librium. The ball rolls down into a valley, usually thelowest part, and then only a strong impact can bring itout again. We say that an atom that has dropped to thelowest level is in a stable state.

But here we must bear in mind that there are alsointermediate states in between the peak and the lowestportion of the valles:. T~e ball may be ~t rest in a slightdepression from whICh It can be ext:1cated by a ~a!tof air, so to speak, or at least by a lIttle push. ThIS ISa metastable state. Thus, besides the excited and stablestates there is a third, metastable, type of energy level.

To summarize, then, the transitions will occur in bothdirections, First one atom and then another atom willmove into a hjgher level. In the next instant, they willfall to a lower level and emit light. But at the very sametime, other atoms will receive energy and will rise toupper levels.

The law of conservation of energy requires that thenumber of transitions upwards equal the number of tran­sitions downwards. What does the number of transitionsupwards depend on? Two factors: first, the number ofatoms in the lowest floor, and, second, the number ofimpacts that raise them to a higher floor. And the numberdownwards? It is of course determined by the number ofatoms lying in the upper floor, and it would seem to be in­dependent of any other factors. That is precisely whattheoretical physicists thought at first, and yet the piecesdidn't fit. The number of transitions upwards, which is de­pendent on two factors, increased with temperature much

Soft Electromagnetic Radiation 19

faster than the number of transitions downwards, whichis dependent on only one factor. This model, which hadappeared to be so obvious, turned out to be nonsense:SOOner or later all the atoms would be chased up to thehighest level: the system of atoms would be in an unstablestate with no radiation.

It was precisely this impossible conclusion that Ein­stein, in 1926, picked up from the reasoning of his pre­decessors. Apparently, there was some other influenceaffecting the transitions of atoms from the upper floor!o the lower floor. One could only conclude that thereIS a forced transition in addition to the spontaneoustransition to the lower level.Wha~ ~s th~s stimulated emission, as it is called? In

short, It IS thIS. A system is in the upper level. It is sep­arated from the lower levl:11 by the difference E -E ==hv. Now, if a photon with energy hv is incident Ionthe system, then it makes the system move down to a lowerlevel. Th.e incident photon is not absorbed in the processbut contmues onwards accompanied by a fresh photonof exactly the same kind generated by the first one.t' Do not seek any logic in this reasoning. It was intui­lon, a guess '" and experiment was to prove it right

or wrong. Using the assumption of stimulated emission. 'he are able to derive a quantitative formula that yields

t e graph of emission as a function of the wavelengthof a heated body. The theory proved to be in brillianta~reement with experiment and so justified the hypoth­eSIS.f It is an exciting thought that the practical conclusionsIr~m the fact of the existence of stimulated emission thate _to the invention of lasers were drawn only many

Years later.

Optical SpectraGenerally speaking, any body is a source of soft elec­

tromagnetic radiation. Using a spectrograph (this is aninstrument whose principal component is a prism or adiffraction grating), light can be decomposed into a spec­trum. The spectrum may turn out to be continuous, band­ed 01' line. The spectra of incandescent solids are verysimilar. For that matter, only a few substances can beheated to incandescence. A real rarity is a glowing liquid.The emission spectra of gases are highly informative.Such are the spectra of rays coming to us from distantstars. The bulk of the information we have concerningthe structure of the universe comes to earth in the formof light rays of stellar matter in a gaseous state.

Under terrestrial conditions, it is easy to obtain emis­sion spectra of atoms. Atoms are made to glow eitherby passing a current through the gas or by heating. Inthis manner we can obtain only the spectra of atomsbut not the spectra of molecules. Before the gas beginsto glow, the molecules break up into atoms. That is whyabsorption spectra are studied if the investigator is in­terested in liquids or solids. In the final analysis, thepicture is determined by the system of energy levels.Transitions up or down yield the same information.Simply, do what is easiest.

Spectra that consist of separate clear-cut lines can beobtained only from a gas or a diluted solution. In thesecond book of this series it was stated that the behaviouror dissolved molecules resembles in many respects thebehaviour of a gas. This also holds true for optical spec­troscopy. Unfortunately, the solvent affects the characterof the spectrum, but if we compare the type of spectraof molecules dissolved in different substances, it is pos­sible to take that effect into account and extract fromthe experiment the fingerprints of the dissolved molecule.

Obtaining a cha t··· .lishing the syste~aco;r~~~~speftruf does not mean estab-for many practical u gy ~ve. s of a molecule. Butan album of inform~i~~oses thiS IS not required. With?f spectral lines and thei~bf:tt sp.~~tra (that is, the listmtensity versus frequency) of ensl le~, o~ the curves ofsubstances, we can hy takin thsome amlly of chemicalsubstance and co~ ar' g. e sp~ctrum of an unknownmaterial from the iJ.b~~g ~he eXI?enmental pattern withvery same way that a crim~te~~m~ the substance in thegerprints he leaves. ma IS etected from the fin-

J ust lat~]y, optical spectral analysis has co . .a competItor: radiospectrosco R' me up agamst~htho~s are .still inferior i~ sensfli~ity~~l~Pt~~:~tros~~PJcar oug the I~ferioritywill most likely not last l:e 0 san~ far sup.en~r to optical methods in the identifi~~t~ut

W q~an~lta~lve analysis of mixtures of substan Ions e on t aim here to acquaint the reader w"th ces.pectra of substances. It will suffic t d' I concretee 0 ISCUSS the pattern

21

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1. Soft Electromagnetic Radiation

Figure 1.3

20Photons and Nuclei

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Niels Bohr (1885-19621- the famous Danish physicis~ who creat­ed the first quantum model of the atom and t~us dIsc~v~red t~e

law of quantization of energy. He was an a~tIve particIpant Indeveloping the princip~es of ~ua~t~m rechamcs. ~e demonstrat­ed the fundamental InapphcabIhty- to ~he mlCroworld - ?fconcepts suitable in describing the he?av~our of macroscOpICbodies. He made a very considerable contrIbutIOn to the theory ofthe structure of the atomic nucleus.

Photons and Nuclei 22 1. Soft Electromagnetic Radiation 23

of energy levels of atoms of hydrogen and t.he fundamentalscheme of energy levels of a free molecule.

Figure 1.3 depicts the system of energy levels of hydro­gen. Note the characteristic thickening of levels as wemove away from the zero line.

Incidentally, the zero in the diagram is not a "real"zero actually. An unexcited atom of hydrogen naturallypossesses some energy. But since spectra exhibit energydifferences, it is convenient to reckon energies from thelower line. Depending on the intensity of the "shock"obtained, the atom can rise to anyone of the "floors",hold on for a moment in the nonequilibrium state andthen, via one of two possible modes (spontaneous emissionor stimulated emission), revert to the lower level.

The resulting spectrum may conveniently be split upinto a number of series. Each series is subordinate to itslower level. In the visible portion of the spectrum wehave the so-called Balmer series. The explanation of thisseries was the first triumph of Niels Bohr's theory ofatomic structure.

Not all energy transitions are of equal probability.The higher the probability of transition, the mOI'O intensethe appropriate line. There are also forbidden transitions.

A great achievement of theoretical physics was theexhaustive interpretation of the spectrum of the hydrogenatom via the solution of the famous equation of quantummechanics derived in 1926 by the Austrian physicistErwin Schrodinger (1887-1961).

Atomic spectra are affected by external fields. Thelines split into several components under the action ofan electric field (the Stark effect) and under the actionof a magnetic field (the Zeeman effect). We will not gointo these exciting phenomena here, but it is necessaryto point out that an understanding of them came onlyafter Samuel Goudsmit and George Uhlenbeck made theassumption that the electron possesses sp'in. How spin

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Figure 1.4

reveals itself in experiment directly was discussed in thethird book of this series.

And finally the last remark regarding the pattern ofenergy levels. We see that the limit which the levels come

11'=3

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1. Soft Elec1romagnetic Radiation 25

up to is marked by the number 13.53. What kind ofnumber is this? This is the ionization potential. If wemultiply the electron charge by the magnitude of thispotential in volts, we obtain the amount of work neededto tear the electron away from the nucleus; in otherwords, this is the work that must be done to destroythe hydrogen atom.

Atomic spectra arise as a result of electron transitions,As soon as we move from atoms to a molecule, we im­mediately have to take into account two more compo­nents of energy. A molecule can rotate. and the atomsof a molecule can perform vibrations with respect to oneanother. All these types of energy can likewise be quan­tized, which means they can have only specific discretevalues. Thus, the energy state of a molecule is describedby the state of its electron cloud (electronic level), thestate of oscillatory motion (vibrational level), and thestate of rotation (rotational level). We thus have to dealwith three kinds of information: the number of the house,the floor, and the flat.

But what is the role of the "floor" and the "flat"?What energy levels have big separations and what levelshave small separations? All these questions are answeredin Figure 1.4. This diagram shows two electronic levelse' and e" (the house numbers). The floors are the vibra­tional levels marked v, and the numbers of the flats are therotational levels marked j. True, this differs somewhatfrom the ordinary numbering of houses and flats, whichis continuous; in dealing with molecular spectra we num­ber the flats on every floor from zero. Thus we see thatthe gaps between rotational levels are the smallest andbetween electronic levels (e' and e") the largest.

Suppose a molecule has the following possible electron­ic levels: 100, 200, 300, ... units of energy, vibrationallevels at 10, 20, 30, ." units of energy, and rotationallevels at 1, 2,i3, ... units of energy; then a molecule on

24

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j=O~jc_f 'IJ..f

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J'- 0 £,1"~1-===:::::::""V'=2- '3----1-

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Photons and Nuclei

Photons and Nuclei 26

the second electronic level, the first vibrational level,and the third rotational level will have an energy of213 units.

Thus, we can give the energy of a molecule as follows:

W = Wei + Wvib + Wrot

The frequency of the emitted or absorbed light will alwayscorrespond to the difference (symbol: ~) of two levels:

1v = h(~Wei + ~Wvib + ~Wrot)

I would like to touch on those transitions that involvea change in only one type of energy. In practice, thisoccurs only in the case of rotational transitions, and whythis is so will soon be seen.

We begin our investigation with the absorption ofelectromagnetic waves of a group of molecules startingwith the longest wavelengths, that is, with the smallestportions of energy kv. Until the magnitude of the energyquantum has become equal to the distance between thetwo closest-lying levels, the molecule will not begin toabsorb. By gradually increasing the frequency, we willreach quanta capable of raising the molecule from one"rotational" step to the next. Experiment shows thatthis occurs in the region of microwaves (the edge of theradio spectrum) or, to put it differently, in the regionof the far infrared spectrum. Wavelengths of the orderof 0.1 to 1 millimetre will be absorbed by the molecules.What we then have is a purely band spectrum.

New things happen when we irradiat.e the substancewith energy quanta sufficiently high to move the moleculefrom one vibrational level to another. However, we willnever attain a purely vibrational spectrum, that is, aseries of transitions under which the number of the rota­tional level is preserved. On the contrar~, t~ansitions

from one vibrational level to another wlll mvolve a

1. Soft Electromagnetic Radiation 27

variety of rotational levels. Say, a transition from thezero (lowest) vibrational level to the first can consistin moving up from the third rotational level to the second,or from the second to the first, and so forth. We thusobtain a vibration-rotation spectrum. We observe it ininfrared light (from 3 to 50 micrometres). All transitionsfrom one vibrational level to another will differ slightlyin energy and will yield a group of very close lines inthe spectrum. In the case of small resolution, these linesmerge into a single band. Each band corresponds to adefinite vibrational transition.

We thus move into a new spectral region, the region ofvisible light where the energy of the quantum becomessufficient to move the molecule from one electronic levelto another. Of course, it is not possible here to obtaineither purely electron tramlitions or electron-vibrationaltransitions. Complex transitions arise in which the ener­gy transition is accompanied by a change in the "house",the "floor", and the "flat". Since a vibrational-rotationaltransition constitutes a band, the spectrum in the visibleregion will be practically continuous.

The characteristic spectra of atoms and molecules havefor many years played the role of helpers in determiningthe chemical structure and composition of substances.And their aid continues today. Revolutionary events inthe field of spectroscopy have only just recently occurred.

Laser Radiation

The first thirty years of this century saw fantasticadvances in theoretical physics with the discovery ofsuch important laws ofnature as the laws of the mechanicsof high velocities, the laws of the structure of the atomicnucleus, and the laws of quantum mechanics. And theforty years that followed exhibited just as phenomenala development in the applications of theory to practice.

~.

I Photons and Nuclei 28

This was a time when humanity harnessed the energyof atomic nuclei and invented semiconductor transistorsthat revolutionized radio engineering and led to the de­velopment of electronic computers and laser technology.These three applications were actually what producedthe modern revolution in science and engineerinO'.

In this section we discuss lasers. First let us give "'somethought to the problem of why, operating via traditionalmethods, we are not able to generate an intense directedbeam of light.

The strongest stream of light collected into a verynarrow beam disperses and loses its energy over smaildistances. It was only in the science fiction of the Russianwriter Aleksei Tolstoi that the hero devises a "hyper­boloid" capable of generating bursts of intense light raysthat can burn and cut materials and carry tremendousenergy over great distances. Of course we know that itis possible to manufacture a concave mirror that cangene::ate a par~llel beam of light. This requires placinga POInt source In the focus of the mirror. But a point isa mathematical abstraction. All right, suppose we havea small source, bigger than a point. But even then, ifwe heat the ball to 6000 degrees (and no known materialcan ~tand ~ore), we obtain a bea~ of light of miserablylow Intensity. And as soon as we Increase the dimensionsof the source, then instead of a parallel beam of rays weobtain a spread-out fan of light "filaments" and theintensity of the ray of the projector begins to rapidlydiminish with distance.

Thus, the first obstacle to creating a strong beam oflight is that atoms emit light in all directions. That'sthe first, but not the last. Atom& and molecules emitlight without agreeing on how they'll do it. The resultis that rays from different atoms set out at differenttimes, totally unmatched in their efforts. The emissionsof different atoms do not agree in phase, and that means

1. Soft Electromagnetic Radiation 29

that rays from different atoms will frequently annihilateone another. This occurs, as you will recall, when thehump of one wave comes with the valley of another one.

Laser emission is what overcomes these obstacles. Theword "laser" stands for light amplification by stimulatedemission of radiation.

The underlying idea is made up of several elements.First of all, recall that there is stimulated emission andspontaneous radiation. We have already mentioned thatthis type of emission occurs when a light photon encoun­ters an excited atom. If the excitation energy of the atomis exactly equal to the energy of the photon, then thephoton de-excites the atom, which moves to a lower leveland emits a photon. The marvelous peculiarity of stim­ulated emission is that the new photon is the sameas the one that generated it, and not only in energybut also in phase and direction of motion.

The second element behind this idea is this. If thesystem of emitting atoms is placed in a tube, the endsof which are at a certain distance from each other andcan serve as mirrors for the photons that interest usthen we can build up a bunch of photons generated (a~they move back and forth) by identically excited atoms.

The third element in this idea is to retain the atomsin the excited state as long as possible and then, after thepumping is completed, to force all atoms to de-exciteat the same time. Putting this laser idea (that is, theproduction of millions upon millions of identical pho­tons from a single photon) in~o hardware should permitgenerating a light beam of ummaginable intensity. Sucha beam would exhibit only the slightest spread andwould have tremendous energy over its cross section.

But the question is: How can this be attained? For"ecades no one knew. Back in the 1930s, importantideas in this connection were expressed by Soviet physi­cist V. A. Fabrikant. Later, persistent research of Soviet

Photons and Nuclei 30

3I- ~:I:

2~::; I-2 :I:

UJIt)

l.lJ ::;0:: i=ll!> U,)

0::

1Figure 1.5

scientists A. M. Prokhorov and N. G. Basov and, inde­pendently, the American physicist Charles Hard Townesled to the invention of lasers. All three received theNobel prize in physics.

Suppose a system has two energy levels. Most atoms ormolecules are in the lower level. Thermal shocks cantransfer a molecule to the upper level for a short time.But not for long, because the molecule is de-excited. Inthe process most of atoms go to the lower level sponta­neously. Of course, some of emitted photons will carrysome of the excited atoms to the lower state and generatephotons of stimulated emission. But these are rare pro­cesses because there are few excited particles (the mostoccupied are the lower levels), and also the probabilityof a spontaneous transition is substantially higher thanthe probability of stimulated emission.

Let us suppose it was possible to find a substance whoseatoms have the three energy levels marked in Figure 1.5by the numerals 1, 2, and 3. The distance 1-3 correspondsto the frequency of emission of green light, the distance1-2 corresponds to the frequency of red light. Now supposethe probability of a transition from level 3 to level 2 isthousands of times higher than the frequency of tru11si­tions from level 2 to level 1. Let us irrajiiate the substancewith green light. The atoms will rist! to the third floor,

1. Soft Electromagnetic Radiation 31

then via spontaneous transitions will go to level 2 and willstay at that level. This is termed nonradiati ve transi tion.The energy released goes into the vibrational energy ofthe atoms. Using our imagination further, let us supposethat we have carried most of the atoms to level 2. Wehave thus reversed the occupancy density, it is no longer"normal". There are more in the upper levels 2 than inthe lower levels 1, which is impossible when the processis controlled solely by thermal motion.

And still and all there doe~ begin a transition fromlevel 2 to the lower level 1. An appropriate photon willencounter other atoms in the excited level 2. The resultwill be not absorption but the creation of a new photon.The first, accidentally generated photon in 2-1 will bejoined by the very same photons of stimulated emission.

Thus arises a stream of 2-1 photons. They will all beidentical and will generate a beam of tremendous inten­sity.

That precisely was the process that the three Nobelprize winners were able to create. Historically, the firstwas a ruby laser. The diagram of levels shown in thefigure is precisely the diagram of ruby with an admixtureof chromium atoms.

To make a laser we need a source of excitation thatdoes the pumping of the laser, that is, that carries theatoms to higher levels.

If the source of laser emission is a solid, then it ismade in the form of a cylinder whose bases play the partof mirrors. In the case of a liquid or gaseous laser, a tubeis constructed with mirrors at the ends of the column.By performing a micrometre-precise positioning of themirrors and thus fixing the length of the column, we putin a privileged position only those photons whose integernumber of wavelengths fit into the length of the column.Only then do all the waves combine.

Perhaps the main peculiarity of the laser is the possi-

Laser emission, that is, the transition to an emptylower level labelled 1, has a wavelength of 1.06 micro­metres.

The dashed line, which depicts the transition fromlevel 1 to the lowest level "does not work" (in the sensethat energy is released in the form of noncoherent radia­tion).

A neodymium laser permits obtaining f.antastic pow~routputs of up to 1012 watts. The energy IS generated III

the form of pulses lasting 0.1. nanosecon~. . .A new competitor in this field IS a laser usmg tranSItIOns

in excited atoms of iodine (Figure 1.7). The workingsubstance here is the gas CsF7I. Here, too, flash bulbsare used for pumping, but the physical proc~sses aredifferent. Ultraviolet light of wavelength 0.25 mlCro~e~reis used for pumping. Under the action of this radIatIOnthere occurs a dissociation of the molecules. The remark­able thing is that the iodine atoms are torn out of t~emolecule and are in an excited statel As the reader WIllsee, this is quite a different method for inverting the

3-2542

o

NONRADIATIVETRANSITIONS----~

Nd

~.L-_.....-2

i.06-pmLASER1

33

t~D:::~ PUMPING BYIII ABSORPTION

Figure 1.6

1. Soft Electromagnetic RadiationPhotons and NucleI 32

biIity of creating a narrow stream of radiation. Practicallyspeaking, a laser beam can have almost any cross section.Technically, this is achieved by the fact that the ray ismade to travel along a narrow glass capillary tube of suf­ficient length. Photons moving at an angle to the capillarydo not participate in the photon build-up. A resonancecavity (that is, the mirrors that reflect photons first inone direction and then in the other during the pumpingperiod in the operation of the laser) reproduces photonsof only OIle direction. In some cases, if an angular disper­sion of the beam of the order of one degree is not satis­factory, a supplementary lens is placed in the path ofthe released ray.

A laser device is a complicated piece of engineering ifone has to do with high power outputs. A primary pulseis first set up in the column; then it is fed to amplifiersthat function in the same manner as the first column butpump independently of the first column. We will notgo into these details because we are only intere~.ted in thephysical principles of pumping and the generation oflaser emission. They can differ greatly, as is evident froma glance at figures 1.6 to 1.8 with diagrams of the actionof lasers which today yield beams of maximum poweroutput.

Figure 1.6 depicts a so-called neodymium laser. Actual­ly, the body of the laser is not the metal neodymium butordinary glass with an admixture of neodymium. Ions ofneodymium atoms are haphazardly distributed among theatoms of silicon and oxygen. Pumping is performed byflash bulbs. The lamps emit in wavelengths between 0.5and 0.9 micrometre, and a broad band of excited statesis obtained (shown here in the form of five bars). Theatoms perform nonradiative transitions to the upper laserlevel (labelled 2 in all three figures). Each transitionyields a different energy, which is converted into thevibrational energy of the whole "lattice" of atoms.

i- .

>­~Il::L1I2\IJ

Photons and Nuclei

Figure 1.7

t

34

Iodine

T'" DISSOCIATION

"'/

PUMPING- B~f.3~~~ABSORPTlO~Y)-i.A

//,

"RECOMBINATIO~,/~

I."OL-..__:...... _

1. Soft Electromagnetic Radiation

Figure 1.8

35

CO"ENERGY TRANSFER

N.2

/--'. 2

I T i~'~;r:PUMPING-

>- BY

~ I ~~ -LoL..--====--------'====____

occupancy density. The operating transition 2-;.1 leadsto laser emission with a wavelength of 1.3 micrometres,after which the iodine atom joins up with the molecularresidue.

The reader has also probably heard of the widespreaduse of helium-neon lasers. They are used to obtain anintense infrared ray of wavelength 1.13 micrometres.These lasers are not record holders as far as power goes, andso we give a diagram of levels for a different laser thatoperates on a mixture of nitrogen and carbon dioxide(Figure 1.8).

Before describing this laser, a natural question comesto mind, and that is: Why are mixtures of gases needed?The point is that some atoms and molecules are moreeasily excited while others are more easily de-excited,so that in a laser operating on a mixture, particles of onetype effect the pumping process, these then transfer theenergy via collisions to other atoms or molecules, andthese in turn generate the laser beam. -There are systemsnow functioning that consist of more than two gases.

For instance, in the nitrogen-carbon dioxide laser, it isadvisable to add a variety of components including he­lium.

Pumping in the CO2 laser is done differently from thetwo just described. The mixture of gases is placed in agas-discharge tube and a sufficiently high voltage is ap­plied so that the system becomes a plasma. Electronsmoving at high speeds excite the vibrations of nitrogenmolecules. The diagram shows a transition of this mole­cule to the upper floor. The voltage applied to the electrodesplays a delicate role. The optimal energy for excitingnitrogen molecules is about 2 eV.

The nitrogen molecule is only an intermediary. It doesnot by itself produce any emission but rather transfersthe energy obtained from the electrons to the CO2 mole­cule and lifts it to an upper laser level.

The upper laser levels 2 are "flats of the third floor"of CO2 molecules. A molecule of gas has a lifetime in theupper laser level equal to about 0.001 second. This isnot so little, and the molecule has a good chance of en-..

~~6

that will force it

37Photons and Nuclei

countering a photon of suitable energydown to a lower level.

It should be pointed out that "interflat" transitionsare a much more frequent occurrence than "interfloor"transitions. Lifetimes on the rotational level are of theorder of ten millionths of a second. This favourable cir­cumstance results in the flats of each floor being occupiedin a rather stable fashion, and so, using an engineeringtechnique that we have already mentioned (setting asuitable distance between the mirrors), it is possible toisolate some one transition; let us say, from the sixthflat of the third floor to the fifth flat of the second floor.

The designing engineer must have at his disposal com­plete information about the residence time of an atom onany given sublevel and about the probabilities of transi­tion. Then he is able to choose the optimal radiationof the given gaseous mixture. Ordinarily, a laser oper­ating on carbon dioxide is tuned to a wavelength of10.5915 micrometres. For a laser to function normally,it is necessary that the molecules not pile up on the lowestlaser level, the idea being for them to do their jobs andget out. Now, at a gas pressure of 1 millimetre of mercury,carbon dioxide molecules experience 100 collisions persecond in vacating the level. The respective figures are4000 and 100000 if helium and water are present. Thisis a tremendous difference.

By selecting the proper admixtures for carbon dioxide,we can boost the power output of the instrument substan­tially. It would appear that this is the gold-medal winnerin the laser field.

A carbon dioxide (C02) laser produces a beam thatcan be focussed on an area of 0.001 cm2 with an intensityof 1000 kW/cm2 in continuous operation and one millionkW/cm2 in pulsed operation with the pulse time equalto one nanosecond (which, as you know, is 10-9 , or onethousand millionth, of a second).

1. Soft Electromagnetic Radiation

The search for suitable materials for lasers is a sortof art. One needs intuition, ingenuity, and memory tocreate an effectively operating laser. The user can noworder lasers with a areat variety of wavelengths rangingfrom a tenth of a mi~rometre to hundreds of micrometres.

The exceptional intensity and coherence of laser emis­sion have revolutionized many areas of engineering. Dur­ing the past decade. the manufacture of la~ers.has becomea whole industry. Lasers have found applIcatIOn as gener­ators of radiati~n that trasmit not only energy but alsoinformation. Intense research is in progress for their usein initiating thermonuclear reactions. Lasers are used inplace of the surgeon's knife in medicine.' as an instrum~nt

for the most delicate surgical operatIOns, as a deVIcefor the separation of isotopes. We will have occasion lateron to come back to further discussions of the marvelouslaser.

Luminiscence

Thermal radiation is a universal property of all bodies.Thermal rays are emitted by every body at any tempera­ture from absolute zero upwards. The thermal spectrumis a continuous one and is depicted by a curve that wehave already discussed. True, our curve was t~at .of .ablack body, but the behaviour of coloured bodI~s IS m.principle but slightly different from the behaVIOur ofblack bodies. Merely, the curve for coloured bodies isdistorted somewhat. But the general increase in the ener­gy of emission (as the temperature rises) and the displace­ment of the maximum to the left (If wavelengths arelaid off on the axis of abscissas) are the general law.

All radiation consists in a transition from a higherenergy level to a lower level. But t~e reasons for excita­tion of atoms or molecules may dIffer. In the case of

Photons and Nuclei 33

thermal radiation, it is the collisions of particles of thesubstance due to thermal motion.

l?ut that is not the only reason compelling a body to~mIt wa,:es. Luminescence, which we are about to discuss,IS o.f a .dIfferent nature. This term embraces processes of~xcItatIO? of molecules that are not connected with anymc~eas~ m the temperature of the body. Causes of particI~excItatIOn may be encounters with beams of photons orelectron.s, mechanical impact, friction, and so on.

Practlcally all substances are capable of luminescence.B~t only some (called luminophors or phosphors) glowbrIghtl! and are of practical importance.

. I:ummophor~ are used as materials for covering tele­VISIOn and OSCIllograph screens, in which case the lumi­nescence occurs under the impact of electrons. Certains';lbstance~ l~minesce brightly under the action of ultra­vIOlet radIatIOn. The energy of the incident photon mustbe at ~east greater than the energy of the emitted photon.That IS 'Yhy. ~he incident quantum of energy can comefrom th~ I~vIsIble portion of the spectrum while the emit­ted ra~IatIOn can lie. in the visible portion.

AdmIxtl;lres of.l';lmmescent material measured in mi­nute fractIOns (b~lhonths, or 10-9) are sufficient to makea. substance !ummesce under irradiation by ultravioletlIght. That IS .why fluorometric analysis is sometimesuse~ as a ~ool III chemical analysis. It is capable of de­tectmg mmute quantities of impurities.

Luminophors are used to cover the walls of daylightlamps.

There are two types of luminescence: fluorescence andp.hosphorescence. Fluorescence consists in the de-excita­tIon of an atom or molecule that occurs without themolecule remaining in the excited level. Contrariwiseph~sphorescence persists after the excitation has ceased:ThIS OCcurs if the system, when excited, passes to ametastable level, from which transitions downwards have

1. Soft Electromagnetic Radiation 39

a low probability. As a rule, the radiation occurs afterthe molecule first absorbs the energy and rises to a higherlevel, after which de-excitation takes place, the transi­tion to the lower level occurring without any stop at anintermediate, metastable, level.

A few words about electro luminescence that occurs incertain semiconductor diodes on the boundary of a p-nregion. This interesting phenomenon is of great practicalvalue because it underlies the manufacture of semicon­ductor lasers. The idea is this: an electron and hole ofthe semiconductor can recombine with the emission ofa photon.

For transitions of this type to take place continuouslywe have to pass an electric current through. the diode.The problem is to find a suitable material that satisfiesseveral requirements. First of all, the current has toinject (if that is the word) electrons into the p-type semi­conductor material, that is, a semiconductor containingmore holes, or it must pump holes into an n-type crystal.This is a necessary condition, but other factors such as,for example, the rate of transition from an upper to alower level can playa decisive role. Then there are caseswhere all factors favour a transition of an electron down­wards and electroluminescence takes place.

A particularly good electroluminescent material is thesemiconductor gallium arsenide. It yields a sufficientquantity of photons. The photons move along the p-nboundary. Two sections of the diode perpendicular tothe boundary are polished and that sets up a resonantcavity. The photons generated in recombinations of holesand electrons are in phase, and for sufficiently largecurrents the radiation becomes stimulated emission withall the resultant consequences of being narrow, highlydirected, and polarized.

Semiconductor lasers operate in a band of wavelengthsfrom the ultraviolet to the far infrared and are widelyused for a variety of purposes.

That most likely was the reason Rene Descartes'(159J3-1650) article entitled Discours .de la Me.thode (1637)was scoffed at by his contemporanes, for It was herethat Descartes would appear to have "proved" this lawwith the aid of rather strange reasoning. Descartes' nebu­lous verbiage did not excite the admiration of his col­leagues. But the fact that his discourse did indeed resultin the correct formula was explained away very simply:he was said to have fitted arguments to a result thathad already been obtained. And so Descartes had alsoto experience accusations of plagiarism.

Perhaps, after all, we might join the sceptical attitudeof his contemporaries. Descartes considers a ball thrownonto a weak net. The ball breaks through the net andloses half its speed. Then, writes. the great p~ilos~P?er,the motion of the ball differs radIcally from Its ongmaldesign, in one direction or another. It is indeed hard toget at what the real meaning is. Perhaps Descartes wantedto say that the horizontal component of velocity of theball does not change while the vertical component does,

2. Optical Instruments

The Prism

The armamentarium of instrllments employed in labor­atories and industry have such a big turnover that ifa scientific worker dropped out of research for a decadeor two he would have to start studying all over again.But today and, most likely, in the distant future hewill again meet his old acquaintances, the prism andthe lens. Let us recall the simple laws that light obeys ininteractions with these transparent materials. Incidental­ly, transparency is a relative notion. For certain electro­magnetic waves, even wood and concrete are transparent.

The laws of interaction of a light ray and a body ca­pable of reflecting and refracting the ray are simpleuntil the wave aspect of the light waves becomes involved.These are the law of reflection (the angle of incidence isequal to the angle of reflection) and the law of refraction oflight. It will be recalled that when a light ray falls on theboundary between two media, it is deflected from its recti­linear path. The angles of incidence i and of refraction rare connected by the relation

sin in=-.­sm r

This law was established, in very careful measurements,by the Dutch physicist Willebrod Snellius or Snell (1591­1626), professor at the University of Leiden. The contents ofhis Course of lectures in which he described the phenomenaof light interacting with transparent bodies was well knownto a small (in those days) circle of European scholars.

2. Optical Instruments

Figure 2.t

41

Photons and Nuclei 42

for it is precisely in that direction that the net obstructsthe motion of the ball.

But let us get back to the law of refraction.The angles i and r are ordinariI y laid off from the

position of the normal as shown in Figure 2.1. The quan­tity n, which is called the index of refraction (or refrac­tive index), depends on the media involved. In order tocompare bodies as to optical properties, it is convenientto set up a table of the indexes of refraction for the caseof an incident ray from the air (or, if one is a pedant,from a vacuum) into a medium. In this case, the angle ofrefraction is always less than the angle of incidence,and hence the refractive index is greater than unity.

The refractive index grows with the density of themedium. For diamond, it is 2.4 and, for ice, it is 1.3.

I'm not going to give a table of refractive indexes,but if I did, I'd have to indicate for what wavelength oflight the data are given. The index of refmction dependson the wavelength. This is an important phenomenonunderlying operation of a number of instrumentsthat resolve electromagnetic radiation into a spectrumand is called dispersion.

If light falls from one medium into a medium of smallerdensity, then a complete internal reflection can occur.In this case, the refractive index is less than unity. Asthe angle of incidence increases, the angle of refractionwill approach 900

• Provided )hat sin r = 1, sin i = nthe light will cease to pa~s-rnto the second medium andwill be reflected in toto at the interface. For water, theangle of total internal reflection is equal to 490

•

The refraction of light by mean~ of a plane plate canbe used to shift a ray to a position parallel to itself.And with the aid of a prism a light ray can even be turnedaround.

If the reader wants to recall the derivation of the for­mula for the angle of rotation D of a light ray, he can

2. Optical Instruments

find it in a school textbook. The derivation only requiresa few facts from elementary geometry, but it is very un­wieldy particularly if it is done for a thick prism andfor an' arbitrary value of the angle of incidence of therayon the prism. A simple formula is obtained if theprism is thin, and the angle of. incidence of the ra~ onthe face of the prism does not dIffer greatly from a nghtangle. If that is the case, then

D=(n-1) p

where p is the angle between the faces of the prism.Using a prism, the great Newton demonstrated for the

first time (this was at the end of the seventeenth century)that white light is not monochromatic but consists of raysof different colours. Violet rays undergo the greatest de­flection and red the smallest. That is precisely why wesay ult;aviolet and infrared rays, and not infraviolet andultrared.

The scientific world learned of Newton's discovery in1672. In explaining his experiments, Newton is clear andexact. Therein lies his genius. As for his discussionsof the matter, it is no easy job to plough through them.Only after much digging in the verbiage can one gatherthat although the author had promised to depict facts andnot to create hypotheses (Newton's famous phrase: "hypo­thesis non fingo", or "I do not frame hypotheses") he didnot carry out his promise. Many of the axioms and defini­tions, like, say, "a ray of light is its minutest part" arevery strange indeed to the mode.rn e?r.

In chemistry the spectrograph stIll reIgns supreme, andthe main component is Newton's prism. The materialmust possess a high degree of dispersion .. Prisms for spec­trographs are made out of quartz, fluonte, and !-,ock salt.The light to be resolved is passed through a slIt locatedin the principal focal plane of. the input l.ens. That iswhy a parallel beam of light falls on the pnsm. Photons

Photons and Nuclei 44

of different frequency go in different directions. The sec­on~, or exit, lens collects identical photons into a singlep~mt of the focal plane. The spectrum may be viewedwIth the naked eye, but then a piece of frosted glass isrequired. The sp~ctrum can also be photographed.

At the present tIme, spectra are registered by automaticrecorders. An ener~y receiver in the form of a photocellor thermocouple slIdes along the spectrum. The receivergene~ates . a cu~rent whose strength is proportional tothe lIght mtenslty. This current deflects the movina partof the recorder in exactly the same way that the c~rrentof a galvanometer deflects the needle. The deflected parthas an attached stylus that records the spectrum on aroll of paper tape that unwinds at a constant rate.

The Lens

A whole industry is engaged in the manufacture of~enses. These are transparent bodies bounded by two spher­Ical surfaces or one spherical surface and one plane surfaceand they come in all imaginable sizes. Some device~use lenses the size of a small coin, in others (large tele­scopes) there are lenses several metres across. The mtlnu­facturmg of large lenses is a real art, because a goodlens must be homogeneous throughout.

Every reader knows wh# a lens is and probably knowsthe main properties of one. A lens magnifies, a lens Canfoc~s rays of light. Using a lens placed in a strong beamof. lIght (from the sun), it is easy to set a piece of paperafire. A lens "collects" rays of light in a single pointthe focus (focal point) of the lens. '

The fact that parallel rays converge to a single pointand, conversely, that a lens produces a parallel beamof rays if a point source of light is placed in the focusof the len~ can be demonstrated with the aid of the lawof refractIOn and simple geometric reasoning.

2. Optical Instruments 45

If a point does not lie in the focus but at a distancea from the centre of the lens, then rays emanating fromit collect at a distance a'. These two distances are con­nected by the familiar formula

1 1 1a+7=Y

where 1 is the focal le~th of tbe~ , (. It IS easy to show that light rays proceeding from an

object twice the focal length produce an inverted andreduced (in the ratio of a' fa) image between the focusand the double focal length.

If the object is placed in the position occupied by theimage, then the image takes up the position occupied bythe object. This is the so-called principle 01 reversibilityof light rays.

When we use a lens to magnify, the object lies betweenthe lens and its focal point. In this case the image is notinverted and lies on the same side as the object.

The difference between the case of a magnifying glassand the two preceding instances is this: a magnifyingglass produces virtual image, whereas in other positionsof the object we obtain images that can be seen on a screenor photographed. We can justly call them real.

The magnifying power of a magnifying glass is thegreater the smaller its focal length. The limiting possibil­ities of a magnifying glass are rather modest: the angleof view at which the virtual image is visible can onlybe magnified 20 to 30 times the angle of view at whichwe see the object with the naked eye.

Many optical instruments would be simple in the ex­treme and would consist of single lenses if it were not fora number of unavoidable defects. We want a parallelbeam of white light to be focussed by a lens to a singlepoint. But dispersion hampers this. Photons of differentcolour are deflected by the lens in different directions.

As a result, instead of a point we obtain a coloured linespread out along the axis of the lens. This is known aschr.Qmatic aberration.

Another bit of trouble is sp..heric;.r!l.!!.!2fl.lIation. The raysthat are closer to the axis of the lens will come to a focusat a more distant point than rays whose paths lie fartheraway from the axis.

Also, the behaviour of rays falling on the surface ofthe lens at large and small angles is quite different. In­stead of a point we obtain a glowing nucleus displacedaway from the proper posiyion. A tail-like appendage isattached to the nucleus. This effect is termed a coma!he word "coma" translated from the Greek is sometTiin~m the nature of "loose hair".

And this ~s not the end of the list of distortions thatplague the smgle lens. When we examine a square we see? quadrangle with the vertices in the form of arcs'convexmw~rds. This is because the rays emanating from thevertICes of the square and from the midpoints of its sidesare refracted in different ways.

Photons and Nuclei

Figure 2.2

46 2. Optical Instruments 47

Another defect of a lens that plagues designers of opti­cal instruments is termed ~stigTn(l.tU;:rt1,.,_Jf a point lies agood distance from the prmclpaI optical axis of the lens,then its image splits into two strips perpendicular toeach other and displaced in opposite directions with re­spect to the position of the ideal image.

And there are still other distortions. Specialists inlens production classify all types of distortions into sevenbasic types. We have mentioned only five.

As so often is the case in technology, making a goodlens amounts to a compromise. It is quite clear· thatincreasing the size of a lens increases the distortionsbut, on the other hand, the illumination of the imag~(that is, the number of photons of visible light per unitarea) is proportional to the square of the diameter ofthe lens (that is, its area). But this is not all. Supposethe object depicted by the lens is at a considerable dis­tance. Then the image will come to a focus. The smallerthe focal length the smaller the dimensions of the image.In other words, the light flux emanating from the objectcollects over a smaller area. Thus, the illumination isinversely proportional to the focal length.

For these two reasons, the square of the ratio of thediameter of a lens to its focal length is called the apertureratio of the lens.­-'ftiick lenses have the smallest focal lengths. These arelenses whose surfaces are formed by small radii. But theseare precisely the lenses that produce the greatest distor­tions. This means that increasing the aperture ratio of alens (whether done at the expense of its dimensions orthe radius of curvature) leads to a worsening of the image.This is no easy task that confronts engineers and design-ers.

Photons and Nuclei 48 2. Optical Instruments 49

The Camera

In its simplest form, a camera is a lens that plays therole of a window in a dark box. The image produced bythe lens is recorded on a photographic plate locatedopposite the window.

Now, a simple lens distorts any image. For this reasonit is replaced by a set of lenses designed to eliminat~the optical defects encountered. The set of lenses istermed a p!y;Qt2.DJ1PltiLfens.

The question now is how to get around the distortionswe have mentioned. Quite some time ago the idea wassuggested to use a system of lenses chosen so that thedefects of each one are eliminated by the defects of theothers .. Thi~J>.!'in.cipl.LQ.Lobtaining a plu~J>.Y- multip!ying

.!-wo mmuses has proved to _be_I?'I!!J!Q[lill-L!()el~m.-iIlata.allseven defects" with the-ala of only thrueiiSes. That ist¥principIe. Actualry~-t<)obtainthe most perfect ima.gesrequires stilT-more complicated combinations. One suchcombination (though by far not the most complicated)is depicted in Figure 2.3. This system of concave andconvex lensesis capable '" 01 prglIllc.ing JLnondi~t.orted'imagtnirider-apprecrable variations of the d~i~i~QJiUag­nificatlOJf:-T~e_.fir,~!_.aD-A.H!iI.d~Q!!!.P.oneDts-.()ftbe system'are~aD1!L,.21 moving 'Y~_th.J'_eslteCJ.t(L,each.other"thusattaining a continuous·-variable three-fold ~gc.~LJl}.!!Kth.

A-Camera requIres 'a raHi,er'slmpTe-ae'vrce-to "focus" it.This means making it poSsible to vary the distance be­tween the centre of the lens and the film. Occasionallyone sees old cameras in the form of an accordion thatcan be squeezed together. And I must say that such came­ras do a fairly decent job.

In a modern camera that fits into the palm of yourhand, this operation is performed more neatly: just atiny spiral turn of the lens mount. As is evident from ourdiscussion of the aperture ratio of a lens, the quality of

Figure 2.3

the image improves if we diminish the eye of the cameraas much as possible. This is done with the aid of a dia­phragm of variable diameter.

We choose the dimensions so that it is as small aspossible but lets in sufficient light to produce a good im­age for the specified exposure.

Did you ever think of why the old photographs takenwhen photography was in its infancy look so stilted,so tense? The explanation is simple: the photographerhad to resort to long exposures and that is why he ex­horted his client not to move: "Hold it, ... ."

The fight to get a good image at minimal exposure iscarried out along two lines. The first is to perfect the pho­tographic lens. This is done not only by suitable selectionof the geometry orthe lenses that make up the compoundleniL ffiaC()mpOumrtelIS"IDmteTilf()fseyei~Trenses nearlyhatftlle light is refl~_cM;~J!l!'~t, this results in a loss ofillum.ination of the image, second, it prodlices a lightnackgrouna-that reduces tlie -'ilegr-ee-()Icolltrast of thefiiiage:-TIi1S1S cOIlloiitlea--1jy --il tec1illIijUe-cal1llif1WScoating. The surface of each lens iscov~!,ecl}Yit.h..~ 'veri"t~. Due to th~~~.o.~fOl.~O_~~~?rinterfe!IlIlc.~L_~he4-2542 .-------

Photons and Nuclei 50

~. 2Q!l..ion of ,.fleeted light is dmtically reduced. Com­

.QQlln~ s~~!~~~ with coated lenses ean"be recognizeaatonce: 1lie glassnas a bI~_sh tinge.

. Another way to improve photographs is to perfect thefilm.

A few words are in order concerning the photochemicalprocess that leads to the formation of an image. Thephotosensitive layer is a gelatin with embedded crystalsof sil vel' bromide and a small admixture of silver iodide.The size of the crystal grains ranges from a thousandthto a ten-thousandth of a millimetre. The number of grainsper square centimetre of film is anywhere from ten thou­sand to hundreds of thousands. Under the microscope,the layer of photographic emulsion reveals that the grainsare rather close together.

Photons falling on a grain of emulsion disrupt thebonds between the atoms of silver and the atoms of thehalide. The number of atoms of silver that are releasedis strictly proportional to the number of photons incidenton the film. The photographer chooses an exposure timeduring which a considerable number of bonds betweenthe atoms of silver and bromine are disrupted. And yet theexposure should not be too long. That would result in acomplete destruction of the bonds between atoms of silverand bromine in all crystals. When the film is developed,the crystals release all the silver they contained andthe film is black throughout. /

If the exposure is correct, the/photographic plate willreveal the latent image of the object. In each grain, thenumber of disrupted bonds is proportional to the numberof photons incident on the grain. The process of devel­oping the film consists in permitting the potentiallyfree atoms of silver to combine. Then the amount ofreleased silver on the negative after developing the filmwill be proJlortional to the intensity of the light.

From the fOl'ogoing it is probably evident to the reader

2. Optical Instruments 51

that the smallest details revealed by a photograph ofan object cannot be larger than the size of a grain crystalof silver bromide.

After tho film is developed the next stage is to fix it.The fixing process consists in removing the undecomposedsilver bromide. If these grains are not removed, thenthe film is spoiled when exposed to light because thegrains release all the silver they contain.

The physics of obtaining a positive image is so obviousthat we will not dwell on it.

The technology of modern coloured photography is notat all simplo and merits our full admiration, yet thephysics of that process, on the contrary, is very simpleindeed. The model of our perception of colour that wasproposed in the middle of the eighteenth century is quitetrue. The human eye has receptors of three colours: red,green, and blue. By combining these colours in differentcombinations, we obtain a perception of any colour.Accordingly, to obtain a coloured image we need a three­layer film. The upper layer is sensitive to blue rays,the middle layer to green, and the bottom layer to red.We will not go into how chemists achieve this. The col­oured negative is transformed into a coloured positive,again through the use of three-layer photographic paper.

The Eye

The eye created by nature is marvelous physical in­strument. The possibilities of distinguishing tens of thou­sands of shades of colour, of seeing close to and far away,of perceiving, via two eyes, the spatial relationships ofobjects, of being sensitive to extremely slight light in­tensities are all properties that place the human eye inthe category of the highest-quality instrument. True,the human eye sees only a small portion of the spectrum.4·

Photons and Nuclei 1)2

The eyes of some animals, however, overcome this defectto a certain extent.

In some ways, the eye is reminescellt of the ordinarycamera. The role of the camera lens is played by thecrystalline lens, which is double-convex. The crystallinelens of the eye is soft and capable of changing its shapeunder the action of muscles that embrace it. Thereinlies the process of accommodation of the eye that permitsit to see with equal ease both nearby and distant objects.With age, the crystalline lens becomes hard and themuscles weaken, and then glasses (spectacles) are neededfor distance and reading.

The image of an object is projected onto the rear wallof the eye, and the optic nerve transmits this perceptionto the brain.

The normal eye of a young person is capable of discern­ing the details of an object located at a distance not lessthan 10 centimetres. Ordinarily, with age comes farsight­edness, and that distance increases to 30 centimetres.

In front of the crystalline lens is the pupil, whichplays the role of the diaphragm of the camera. The humanpupil can vary in size from 1.8 to 10 millimetres.

The role of the photographic plate on which the imageis formed is played in the human eye by the retina, whichhas a very complicated structure. Under the retina liesthe optic epithelium, which consists of light-sensitivecells called rods and cones. You can compare the numberof these cells to the number of~ains of silver bromidein the photographic plate. l'1iere are over a hundredmillion optic cells in the human eye. Since the normalperson is capable of distinguis?ing co~o.u~s, it i~ clearthat the optic cells possess differmg senSItiVIty to dIfferentportions of the spectrum. We arrive at the same resultif we assume that the cells are divided into classes recep­tive to different portions of the spectrum.

If vision is normal, the rear focus of the eye in the

2. Optical Instruments 53

calm state lies on the retina. If it lies in front of the retina,the person is nearsighted; if it is behind the retina, theperson is farsighted. These two most common defects ofsight are due to an excessively thick or thin crystallinelens. Some people also suffer from astigmatism. In thiscase, the crystalline lens in the normal state does nothave a correct shape bounded by two spherical surfaces.

All these defects can be rectified by eye glasses, which,together with the crystalline lens, yield an optical systemthat can focus the image of a distant object on the retina.

The lenses of spectacles are characterized by what iscalled diopter (a unit of the power of lenses). The opticpower of a lens is inversely proportional to the focallength. The optic power in diopters is equal to unitydivided by the focal length in metres. The focal lengthsof dispersing lenses that nearsighted people use in theireye glasses are negative.

The angle of vision of the eye is much greater thanappears to us. A number of events that occur at rightangles (90°) to either side of our straightforward vieware recorded directly by our subconscious. This fact leadssome people to the erroneous idea that they "feel" some­one is looking at them, without actually seeing theperson.

The eye poorly perceives objects seen at an angle ofless than one minute of arc, even if the illumination isvery good.

Polarizers

A light wave is an electromagnetic wave. As was men­tioned in the third book of this series, pictorial experi­ments have demonstrated that the vector of the electricfield is perpendicular to the direction of motion of a rayof light. If this fact is interpreted from the standpoint oflight being corpuscular (in the form of particles), then

Photons and Nuclei 54

we must say that a particle of light-the photon-is nota sphere but an arrow. In a number of complicated cal­culations, theoretical physicists have come to the con­clusion that the photon possesses spin (equal to 1). Thus,the concept of a photon as an arrow is quite natural.

The ordinary ray of light is a stream of photons whosespins are perpendicular to the direction of propagation ofthe light but are distributed uniformly in the form of acircle perpendicular to the light ray. This type of lightray is said to be nonpolarized. However, in a numberof cases we have to do with a beam of photons in whichall the spins are in one direction, or, to put it differently,we have to do with electromagnetic waves, the electricvectors of which have a very definite direction. Suchrays are said to be polarized.

One way to obtain polarized rays consists in makingthe light ray pass through a low-symmetric crystal. Suchcrystals, when suitably oriented with respect to an inci­dent light ray, possess the capability of splitting thenatural ray into two rays polarized in two mutuallyperpendicular directions.

Unfortunately, I cannot give the reader even a roughidea about how this happens because the molecules ofa crystal "receive" waves with differently arranged elec­tric vectors in different ways. I can see that the precedingsentence hasn't helped matters much. But I can at leastassure the reader that the theory of the splitting of lightrays exists and it is a very good theory that is capableof describing all the fine details of this exciting phenom­enon. For instance, it is possible to predict how thepassage of light will change if we place the crystal atdifferent angles to thwray of light.

By splitting a nonpolarized ray into two polarized rays,we can then easily make one of the rays go orf in somedesirable direction. The result will be what is called aNicol prism, named after the Scottish physicist William

2. Optical Instruments 55

Nicol (1768-1851). The instrument was made in '1828.I t is interesting to note that in those days all explana­tions of the polarization of light were given in the lan­guage of particles and it was considered an excellentconfirmation of the corpuscular theory of light proposedby Newton.

Soon afterwards the interference and diffraction of lightwere discovered, and these were so naturally accountedfor in terms of waves that the theory of particles of lightwas buried. But a century passed and the theory wasresurrected, like the phoenix arisen from ashes. Truethis time merely in the modest attire of one of two as~pects of the electromagnetic field.

If a polarizer is placed in the path of the light, theintensity of the ray falls off, as is to be expected, by afactor of two. But the most interesting phenomenon,which is what proves the existence of polarization, occurswhen we place a second instrument in the path of thelight ray. This instrument is called an analyzer, althoughit does not at all differ from the first Nicol prism. Nowlet us turn the Nicol prism about the light ray. It turnsout that for a certain mutual position of the two prisms,the intensity of the light that has passed through bothNicol prisms remains the same as it was in the absenceof prisms. We then say that the Nicol prisms are parallelin that position. Now start turning the analyzer. Whenwe have turned it through 90 0

, the light ceases to comethrough. We then say the Nicol prisms are crossed.

In an intermediate position when the second Nicolprism is turned from the parallel position through anangle U, the intensity is equal to (1/2) I cos2 u. The formulais readily explainable if we assume that the vector ofthe electric field has been divided into two components,one perpendicular and the other parallel to the "slit"of the analyzer. Now the intensity is proportional to theftlquare of the amplitude of the wave, that is, to the square

Photons and Nuclei 56

of the electric vector. That is why variation in lightintensity must occur in accord with the law of the squareof the cosine.

Such an analysis of polarized light has a number ofpractical applications. Suppose the Nicol prisms arecrossed and a transparent body that is capable of turningthe electric vector of the wave is interposed betweenthem. We then have field brightening. Charged bodieshave this capability. Depending on the amount of volt­age, the rotation of the light vector and, together withit, field brightening beyond the crossed Nicol prisms willbe different. We will see pretty patterns, and colouredtoo, because photons of different colours behave differ­ently. These patterns permit judging the voltages inthe sample or deciding whether the molecules that makeup the sample are oriented or not. This is importantinformation, and so a good microscope is equipped withtwo Nicol prisms so that the image of an object can beseen in polarized light. The information about the struc­ture is then much richer.

Solutions of many substances (for example, sugar solu­tions) have a certain rotatory power that permits turningthe electric vector of a light wave. Here, the angle ofrotation turns out to be strictly proportional to theamount of sugar in the solution. We can thus adapt a polar­imeter for measuring sugar content. Such instrumentsare termed saccharimeters and can be found in anychemical laboratory.

These two examples do not exhaust the use of polar­imeters but they u~re probably the main ones.

The Microscope and the Telescope

The optical part of a microscope consists of an eyepieceand an objective. The eyepiece (or ocular) is the lens~Q which the e¥~ is applied; the objective isclQse to the

2. Optical Instruments 5',

object being studied. The object is placed at a distancesomewhat greater than the focal length of the objective.The image obtained between the objective and the eye­piece is inverted. It is necessary that it appear betweenthe eyepiece and the focus of the eyepiece. The eyepieceplays the role of a magnifying glass. It can be demonstrat­ed that the magnifying power of a microscope is equalto the product of the magnifications of the eyepiece andobjective taken separately.

At first glance it might appear that a microscope canbe used to discern arbitrarily small details of an object.For example, why not make a photograph by increasingthe dimensions thousands of times, then examine it ina microscope and obtain a million-fold magnification,and so on.

This kind of argument is tl;)tally fallacious. First ofall, let us recall that increasing the size of photographicpictures is limited by the size of the grains of the film.The point is that each tiny crystal of silver bromide actsas a whole. The reader most likely has seen highly mag­nified photographs and has noticed that the magnifica­tion does not at all improve the details of the picturebut only smears them out.

But if we discard the operation of photography andmagnify the image optically, and this is possible merelyby increasing the number of lenses, we will soon see thatany great magnification is meaningless. The useful mag­nification of any instrument is limited by the wave aspectof the electromagnetic field. Whether we are examiningan object through a magnifying glass with the nakedeye or with the aid of a microscope or telescope, in allcases the light wave proceeding from the glowing pointmust pass through an opening. But in that case we haveto deal with the phenomenon of diffraction, that is, thedeviation of tho light ray from its rectilinear path. Toone degree or another, the ray "peeks around the corner"

tories. It is called an electron microscope. The wavelengthsof electrons can be made very small (see page 129).Electron microscopes can resolve structural details ofthe order of ten millionths of a millimetre.

Biologists have seen molecules of DNA, the long mole­cules that carry hereditary characteristics from parentsto progeny. Protein molecules have been observed, alsothe structure of cell membranes and details in the struc­ture of muscle fibres. The photograph in Figure 2.4 isa record: this is a 3-million-fold magnification of thecrystal lattice of the mineral pyrophyllite showing +'hedistance between the planes of the crystal at 4.45 ang­stroms!

The limit of the electron microscope has to do notwith its resolving power (we can easily reduce the wave­lengths of electrons) but with the contrast in the image:the molecule under investigation has to be backed: placedon a base (backing), which itself consists of molecules.On the background of molecules of the backing, it isdifficult to discern the molecule of interest.

An electron microscope is a complex and very expensiveinstrument about a metre and a half in height. Electronsare accelerated by a high voltage. The. principle of mag­nification is the same as in the optical microscope. It isproduced by lenses. But, naturally, they are not at alllike those of an ordinary microscope. The electrons are

Photons and Nuclei 58

and so the image of the point is never a point but a spot.And no matter how you try, it is impossible to makethe size of the spot less than the wavelength of the light.

It is essential to be able to estimate under what condi­tions the path of the electromagnetic wave departs appre­ciably from the rectilinear path.

If x is the linear deviation from the rectilinear pathobserved at a distance f from the source of radiation.and the size of the obstacle (or aperture) in the path ofthe ray is equal to a, then we have the following rela­tionship:

ictx=­a

Here, A. is the wavelength. From this equation it followsthat diffraction can be observed both in the case of mi­nute particles and celestial bodies. The whole point isthe wavelength and the distances we are dealing with.The same goes for apertures as well. Diffraction is notonly observed in the case of minute openings. For exam­ple, an opening the size of a tennis ball permits observingdiffraction phenomena; true, only at distances of theorder of hundreds of metres.

The simple equation we wrote down enables us to gaugethe limiting possibilities of microscopes and telescopes.

A microscope does not permit us to see features of anobject smaller than a micrometre. Now millimetre sizescan be seen with the naked eye. This means that opticalmicroscopes should not be used to seek magnificationsgreater than 1OQO.

This restriction has to do with optical microscopesonly. Now if it were possible to design a microscope oper­ating not with light rays but with some other rays withsmaller wavelengths, then the useful magnifying powerof the microscope could be increased. Just such a micro­scope exists and can be found in many scientific labora-

2. Optical Instruments

Figure 2.4

59

Photons and Nuclei 60

focusse.d with th~ aid of electric fields of high-voltagemetallIc plates wIth apertures and by means of coils thatset up a magnetic field.

Th~re are a great variety of techniques that help toproduce the image. Microtomes are used to obtain ex­tremely thin sections for transillumination, the moleculeson the backing are tinged by depositing metallic vapour onthem. It is also possible to obtain a replica of the speci­men, that is, by coating it with an extremely thin filmof transparent material and then etching out the objectitself .

Electron microscopy is a big and important division ofphysics well deserving of. a separate chapter, but the sizeof this book presses me onwards.

Our next topic is the telescope. As early as the 16thcentury suggestions were made that distant objects couldbe examined with the aid of convex glasses. Yet I thinkthat the real invention of the telescope (actually a spy­glass) was made by the great Galileo. He constructed itin July 1609 and a year later published his first obser­vations of the night sky.

Like the ~ic:oscope,.a telescope. (refracting telescope,to be exact) IS m prmcIple a combmation of two lenses:the objective, that forms an image from a distant objectand the eyepiece. Since an infinitely distant object i~viewed, its image is created in the focal plane of the objec­tive. The focal plane of the eyepiece coincides with theplane of the objective, and beams of parallel rays emergefrom the~yepiece.

The power of the telescope increases with the diameterof th~ obj~cti~e. For example, a 10-cm-diameter telescopepermIts vIewmg features on Mars about 5 km in size.With a 5-metre diameter telescope, objects up to 100 me­tres across can be resolved on Mars.

Astr?nomical observatories are equipped not only withrefractmg telescopes but also reflecting telescopes. Since

2. Optical Instruments 6f

our aim is the study of distant objects and it is requiredto gather the light rays into a focus, this can be done bymeans of a spherical mirror as well as a spherical lens.The obvious advantage in this case (a mirror) is thatwe get rid of the chromatic aberration. The defects ofthe mirror telescope are connected only with the difficultyof making high-quality mirrors.

Quite naturally, the telescope too has a limit to theuseful magnifying power it is capable of producing. Thislimit is associated with the wave nature of light. A ray oflight from a distant star spreads out into a circle andthis places a limit on the angular distance between starsthat we are able to resolve in the telescope. Again, thedesire to increase the power of the telescope is linked upwith increasing the diameter of the glass. The limitingpossibilities of the telescope probably lie somewhere closeto one tenth of a second of arc. .

During recent years, new instruments have appearedand are helping astronomers study the stars in all portionsof the spectrum of electromagnetic waves coming to usfrom outer space. We will come back to this topic againin the seventh chapter.

Interferometers

As we have already pointed out a number of times,the electromagnetic field possesses a wave aspect. And soalso do beams of particles-electrons, neutrons, protons.Sound is the result of mechanical displacements of themedium and these occur in accord with the wave law.Common to all these physical processes is the possibilityof ascribing to any radiation a wavelength, frequency,and rate of propagation; these are connected by the equa­tion c = AV. The simplest kind of radiation is monochro­matic, that is, it is described by a single wavelength. Inthe general case, radiation constitutes a complex spec-

Photons and Nuclei 62

trum, that is, a sum of waves of different wavelength andintensity.

The wave aspect of radiations is revealed in the combin­ing of waves and also in scattering due to bodies in thepath of the rays. An important special case of wave scatter­ing is diffraction. The composition of waves is interference.

Here we will deal with the interference of light. Thisphenomenon underlies the operation of a number of in­struments used for measuring distances and also certainother physical quantities. Instruments that use the phe­nomenon of interference for applied purposes are calledinterferometers.

The principle for measuring distance consists in cal­culating the number of waves that fit into the distancebeing measured.

This might appear to be a simple thing to do. Take twosources of light and bring both beams to a single point.Then depending on whether the waves arrive at the pointof observation "hump to hump" or "hump to dip", weget a bright or a dark spot. Now let us pose the problemof measuring the distance over which we wish to move oneof the light sources. This motion'will cause the phase rela­tionships of the two waves at the point of observation tochange. All we have to do is calculate the number of alter­nations of light and darkness and then, taking into accountthe geometry of the experiment and knowing the wave­length of the light, readily compute the amount of dis­placement.

In principle that is correct. But acting in this fashionwe will not observe a pattern of alternating light anddarkness. The screen will remain light all the time andso this simple experiment is a failure.

What is definitely known is that if two rays of lightemitted by two different sources come to a point, theywill always reinforce each other. Then perhaps the wavetheory is fallacious?

2. Optical Instruments 63

No, the theory is correct and electromagnetic radiationhas a wave aspect. But we began with an erroneoussupposition. For interference to be observed, it is neces­sary that an invariable phase difference obtain all thetime between the waves being combined. But phase re­lationships even between waves emanating from two atomsof one and the same source are quite accidental. We havealready mentioned the fact that atoms throw out photonswithout "agreeing" among themselves on their behaviour.Thus, two different sources radiate haphazardly; this iscalled incoherent radiation.

But then coherent radiation must be something in thenature of a dream. We shall now see that that is not so.

The solution is extremely beautiful and at the sametime simplicity itself, like most original concepts: theproblem is to make the radiation of the atom combinewith itself. And to do that it is required to split the raycoming from each source into two parts and make thetwo parts of the single ray traverse different paths, andfinally bring them together. Then, under such conditionswhile observing in terference and changing the differencesin the paths of the parts of the split ray, we can indeedmeasure the displacement arid length by computing thenumber of alternations of light and darkness.

What we have just described is the principle underlyinginterferometer measurements that was discovered in 1815by the French physicist Augustin Jean Fresnel (1788-1827).Let us now consider the methods underlying the actionof interferometers by means of which one can split aray of light and create different paths between the splitportions of the ray.

Now let us dwell in more detail on the interference oflight rays reflected from the inner and outer sides of atransparent plate or film. This is of interest both becauseof its practical utility and because it is observed in nature.Besides, this is a good example for illustrating many

important concepts that we use in describing light wavesand other electromagnetic waves.

Figure 2.5 enables us to compute the phase shift be­tween two such rays. The difference in phase is determinedby the difference in routes, that is, the difference in thepaths taken by the two rays. From the drawing it is evi­dent that the path difference is x=2d cos r. But thenthere is the question of how to proceed from the pathdifference of the rays to the phase difference, which iswhat determines whether the two waves will reinforceor weaken each other.

If the reader is not afraid of the cosine formula, we candiscuss this matter. The oscillation of a light vector atany point in space can be written down as follows:A cos 21tvt. The phase shift through the angle cp signifiesthe necessity of adding this angle to the argument of thecosine. If we want to compare the phases of points ofone and the same wave, which points are separated bya distance x, we ha ve to take into account the number of

Photons and Nuclei

Figure 2.5

64 2. Optical Instruments 65

wavelengths that fit into that interval and then multiplythe result by 21t. That quantity is the phase shift. Thus,cp = 21tx/A.. . .

Now let us return to the mterference of rays III theplate. We have written down. ~he difference ~n paths.All that is left, then, is to divide that quantity by A.But wait a minute. Is the wavelength of light in a vacuumthe same as inside a transparent plate? Something makesus suspect that when a wave of light passes fl,om onemedium into another it undergoes a change. We knowabout dispersion-when photons of different frequencybehave differently. The frequency, wavelength, and rateof propagation are related as follows: C=VA. Which ofthese quantities undergoes a change when the wave entersa new medium? Experiment can tell us. We can measurethe rate of propagation of the wave directly in the bodyand then see that the refractive index, which makes thowave alter its direction of motion when falling obliquelyon the interface between the two media, is equal to theratio of the rates of propagation of light in the media.If one of the media is air (a vacuum, to be more exact),then

cn=-

IJ

where C is the velocity of light in a vacuum and v is therate of propagation in a medium. Now the question iswhich of the two parameters (frequency or wavelength)undergoes a change when light moves from the air intothe medium. To account for the results of the interferenceexperiments, we have to assume that the frequency ofthe photon remains unchang~d and the wavelengthchanges. That is why the followmg formula holds true forthe refractive index:

1.0n=ywhere 1...0 is the wavelength in air.

5-2542

5*

This method is widely used in optical work. For exam­ple, if we need to check the quality of the surface of aglass plate, this is done by considering the bands of equalthickness of a wedge of air set up by the plate with a perfect­ly smooth surface. If the two plates are pressed together atone end, a wedge of air is produced. If both surfaces areplane, the lines of equal thickness will be parallel lines.

Suppose the plate being tested has a depression or abump. The lines of equal thickness will be bent becausethey bend around any defective spot. As the angle ofincidence of light is changed, the bands move to one sideor the other depending on whether the defect is a bumpor a depression. Figure 2.6 illustrates what the micro­scopic field looks like in such cases. Both patterns revealdefects in the samples. In the first case, the defect islocated on the right edge, and in the other, it is on theleft edge.

Exact measurements of refractive indices of a substancecan be made with the aid of interference refractometers.These instruments reveal interference between two raysthat have the largest possible separation.

Suppose in the path of one of the rays we have a body

Photons and Nuclei 66

Now we know all we need to in order to write down thep~ase difference between the rays in the plate experiment.Slllce one of the light rays moved in the air and the otherin glass, the phase difference is

21t 41t<p = --::;:-nx = -"nd cos r

o "'0

What can we measure by studying the interference oflight rays in a plate? The formula answers that question.If we know the thickness, we can determine the refractiveindex of the material. If we know n, we can find thethickness to a high degree of accuracy (fractions of awavelength of the light) and, finally, we can measure thewavelengths of different colours.

If the plate is of variable thickness and if the materialis homogeneous, and t~e angle of incidence is practicallythe same for the portiOn of the plate we are interestedin, then the interference will be exhibited as so-calledbands of equal thickness. On an uneven plate, we havea system of dark and light bands (they may be all thecolours of the rainbow in the case of white light becausethe photons of each colour will behave in their own distinc­tive way), the bands revealing places of equal thicknessThat is the explanation of the varicoloured streaks that~ see on films of oil spilt onto water.

Soap films exhibit very pretty bands of equal thickness.Make. a simp}e wire frame, dip it into sudsy water, andpull It out. fhe soap suds slide off and the film in theupper part is thinner than it is in the lower portion. Thefilm exhibits coloured horizontal bands.

Interference methods are widely employed in measuringsmall distances or small changes of distance. They permitmeasuring changes in thickness less than hundredths ofthe wavelength of the light. In interference measurementsof uneven places on the surface of a crystal, precision ofthe order of 10-7 centimetre has been attained.

2. Optical Instruments

Figure 2.6

67

that the speed of light is not combined with the speedof motion of the lamp (that produces the light flash) inaccord with the rules by which the speed of a bullet iscombined with the speed of the man holding the gun.The discovery of this remarkable fact led to the theoryof relativity and to a fundamental reconsideration of themeaning of such basic scientific concepts as length, time,mass, and energy. We will go into these important mat­ters later on. The Michelson interferometer is of interestnot only because of the place it occupies in the historyof physics but also because the simple principles under­lying its construction are still used today to measurelengths and distances.

In this instrument, a parallel beam of monochromaticlight falls on a plane-parallel plate PI (Figure 2.7) cov­ered with a translucent layer of silver on the ruled side.This plate is placed at an angle of 45 0 to the light rayfalling from the source and splits it in two, one ray movingparallel to the incident ray (to mirror jlJ1 ) and the otherperpendicular to mirror M2 • The separated rays of lightfall on the two mirrors at zero angles of incidence a-ndreturn to the very same spots of the translucent platefrom which they emerged. Each ray returning from the

Photons and Nuclei 68

of length l and refractive index n. If the refractive indexof the medium is no, the optical difference of the pathchanges by /). = l (n-no)' The two rays are brouO'ht to asingle point by means of a focussing lens. Wllat willwe see In telescope? A system of light and dark bands. Butthese are not bauds of equal thickness that can be seen:vith the unaided .eye. Th~ system of bands appearingm a refractometer IS of a dIfferent origin. This is becausetl~e origi~al beam o~ light is not ideally parallel butslIghtly dI:ergent whIch means the light rays constitutingthe con.e wIll fall on the plane at slightly different angles.

The mterference events will occur in the same mannerin the case of rays having the same inclination. And theywill come together in the same spot of the focal planeof t~e telescope. If the path difference between the splitportIOns of the beam changes, the bands will begin tomove. If the path difference changes by the amount /).then /)./'" bands will pass through the eyepiece of th~telescope.

The method is extremely precise because a displacementof 0.1 band can be detected with ease. For such a displace­ment,. Ll=0.1A=0.5xlO-5 centimetre, and this permitsdetectmg, ?ver.a length l=10 centimetres, a change inthe refractIve mdex amounting to 0.5 X 10-6 •

We will now examine an interferometer of a differentkind that does not utilize the phenomenon of refraction.This interferometer was made by the American physicistAlbert. Abraham Michelson (1852-1931). It is hard tooverestImate the role this instrument played in the histo­ry of physics (I make bold to say, even the history ofhuman thought). Through the use of this interferometera fact of exceptional importance was discovered: th~speed of light in directions along the earth's orbit andacross it is the same.

What this means is that the speed of light does notdepend on the motion of the light sourceI This means

2. Optical Instruments

Figure 2.7

*

69

Photons and Nuclei 70

mirror is again split on the plate. Part of the light returnsto the ~ource and the other part enters the telescope.In. the figure .you can see that the ray coming from themIrror oppos~te .the telescope passes twice through theglass plate wIth Its translucent layer. For this reason, inorde.r to ensur~ equality of the optical paths, the raycommg from mIrror M 1 is passed through a compensatingplate Pz that is identical to the first plate but without thetranslucent layer.

In the field of view of the telescope, circular ringsare ?bserved ~hat correspond to interference in the layerof all' (the thIckness of which is equal to the differencebetween the distances of the mirrors from the site ofbea~-splitting) of the primary rays forming the cone.A dlsplac~~ent of one of the mirrors (for instance, M zto the posItIOn shown by the dashed line) by a fourth ofa wavelength will correspond to a transition from maxi­mum to minimum, that is, it will cause a shift in thepattern through half a circle. This can be clearly detectedby an observer. Thus, in violet rays the sensitivity of aninterferometer is better than 1000 A (angstroms).

The advent of lasers produced a revolution in thetechnique of interferometry.

Did you--ever wonder why an interference rainbow ap­pears. on films of oil floating on water but does not appearI~I 'YlIldo~ glass? The situation would seem to be quiteSImIlar: III both cases, rays are reflected from an outerand an inner surface of a plate. In both cases, the tworeflected rays c~mbi~e. ~ut interfer~nce is observed onlywh~n the plate IS thIll. I ve used thIS question at exami­natIOns and confused many a student.

The essence of the matter is this. The emission time ofan ~t~m is equal to 10-8 or 10-9 second. The single act ofe~IsslOn consists in the release of a wave train. Since thetIme of ClJ.lission is very small, the wave train is veryshort despIte the great velocity of light. When we split

2. Optical Instruments 71

a ray into parts, only two parts of the same wave traincan undergo interference. This means that one portionof the sine curve must appreciably overlap another por­tion. But for this to happen, it is naturally necessarythat the path difference between the split portions ofthe ray be substantially less than the length of the wavetrain. The condition is met in the case of a micron-thinfilm, but not in the case of window glass. That is theanswer the student should have given.

The maximum path difference between the rays wheninterference can be observed is termed the coherence length.For light, this amounts to a fraction of a millimetre.

Now notice how radically the situation changes in thecase of laser emission. A continuous-action laser createsphotons of stimulated emission that set out in the samephase. Or, to use wave language, the wave trains ema­nating from different atoms are superimposed on oneanother thus creating what appears to be a single wave.The coherence length becomes practically unbounded; atany rate, it is measured in metres and kilometres (as isalways the case, the ideal is never attainable but I willnot dwell here on the variety of factors that affect thecoherence length).

Using laser light, it is possible to construct interfer­ometers capable of handling problems that until recentlyhad appeared unsolvable. For example, using an ordinarylight source, the mirror in the Michelson interferometercan be displaced only distances of the order of a milli­metre. Now if the light ray is produced by a laser, thepath of the ray falling on mirror M 1 may be made equalto several centimetres and the path of the ray reflectedfrom mirror M z , tens of metres.

Interferometers to check the sphericity of lenses canbe made with a single surface of comparison whereas,if we use ordinary light, one has to change the standardof comparison along with any change in the radius of

Photons and Nuclei 72

the lens being tested (this is because one cannot workwith large path differences). Which is to say nothingabout the fact that the interference patterns have becomeincomparably brighter and for that reason can be ana­lyzed more easily and exactly.

The possibility of getting along without compensationof the optical path of one of the rays makes it possibleto design interferometers of an entirely new type. It hasnow become possible to detect displacements of dams,geological drift, oscillations of the earth's crust. By re­flecting laser light from objects located at large distancesand then making it interfere with the original light, wecan carry out exact measurements of the speed of motionof such objects.

laser Devices

A device that produces a laser beam can of course becalled an instrument because it is used for analysis,control, and observations. However, unlike other opticalinstruments, the laser plays a far greater role in industry.The use of lasers is so widespread that we will again andagain come back to them. In this section we will discussthe use of lasers in working various materials. If it isnot high power that is required, use can be made of thecompact neodymium laser. The heart of this laser is,as we have already mentioned, glass alloyed with neody­mium. A glass rod 50 millimetres long and 4 millimetresin diameter. The light flash that does the pumping isgiven by a xenon lamp. In order to cut losses of luminousenergy, the lamp and rod are enclosed in a water-cooledcylindrical chamber.

The following properties are important in the use ofthis type of instrument and its analogues: the possibilityof localizing energy over an extremely small area, thepossibility of precise batehing of portions of energy, and

2. Optical Instruments 73

the possibility of delivering energy without the use ofany wires or contacts.

Take the use of lasers in the watch industry. A watchrequires jewels. These are often in the form of rubies;the more rubies the higher the quality of the watch.Apertures have to be drilled in tiny ruby discs. Ordinarily,without lasers, this operation takes several minutes foreach jewel. Using lasers, the process is completely auto­matic and takes a fraction of a second. And since industryuses many millions of these a year, the obvious gain intime is tremendous.

The laser is extremely important also in the diamondindustry. Drilling and broaching of diamonds is easilyhandled by lasers which can make a diamond into anyshape or produce apertures as small as a few micrometres.

But let's get back to watches. A laser can weld thespring to the watch mechanism. Quite obviously, wher­ever high-precision welding is required -and this is theorder of the day in many areas of high-precision technol­ogy-laser beams are employed with excellent results.The great advantage of this needle-thin ray is that thereis no need to protect and cool adjacent parts of the com­ponent being welded.

Lasers are of course used trivially as a knife to cutout any imaginable patterns on any kind of material.

Lasers have invaded some rare professions as well.Take the restoration of marble sculptures. The atmosphereof the twentieth century is unfortunately quite pollut­ed. A great variety of noxious gases, particularly sulphuroxide, form a black crust on marble. This crnst is porousand acts as a sponge packing up more moisture and moreharmful substances. Removing the crust by mechanicalor chemical means can often ruin the sculpture. Now alaser operating in a pulsed regime can remove tho crustwithout harming the underlying marble.

A carbon dioxide laser can be used to grow crystals

75

V(J..)

1.0 (\0.8

I \0.6

'\0:1\0.2 J

~ '-°400 500 600 700

l\-. NANOME:TRES

Figure 2.8

relative units) the sensitivity of the normal. human. eyeto waves of different length. However, engllleers ~Isre­gard this curve and leave it u.p to the h,uman ~ye t? Judgethe integral luminous intensLty. The first thlllg, III thatcase is to choose a standard light source and then compareothe~ sources with the standard. For a long time, theunit of luminous intensity was the candle because atfirst attempts were made to find a c~rt~in standard. candleflame. It is quite obvious that thIS ~s no ~asy Job.

The international standard today IS an lllcandescentblack body. The material is platinum. The black bod,Yemits light radiated through a small aperture by platI­num heated to the melting point, or 1769 DC.

The unit of luminous intensity is the candela (which isLatin for "candle"). The international definitio.1l avoidsany direct indication as to temper~ture of. lumlllesce~lce(this is doue to avoid errors aSS?CIate~ WIth meas.urmgthe temperature). The candela I~ defI?e,(~, thus: If wetake, as the source, platinum as It sohdr,fies a!5 normalatmospheric pressure, then an area of (1IG) X 10 square

2. Optical Instruments

Photometry

Every source of light may be characterized by theenergy it emits. However, in many cases we are onlyinterested in that part of the energy flux that produce'sa visual sensation. As we have already mentioned, thisis characteristic of electromagnetic waves of wavelengthbetween roughly 380 and 780 nanometres.

Light perceived by our brain is characterized by lumi­nance (brightness) and colour. If we compare the visualsensations created by light of equal intensity bnt differentwavelength, it will he apparent that the eye perceivesas hrightest light that has a wavelength of 555 nanome­tres. This is green.

The perception of light is shown as a visibility (orluminosity) curve in Figure 2.8, which represents (in

Photons and Nuclei 74

without resorting to the crucible. This is not a new pro­cedure. High-frequency currents have long been used inthis capacity, but not for dielectric materials that possesstoo Iowa thermal conductivity. Today, lasers are used togrow crystals (without using crucibles) of niobates andother much needed materials. The importance of thenoncrucible growth of crystals for the needs of microelec­tronics cannot be overestimated because even a few partsin a million of impurities can playa negative role, andin a crucible it is practically impossible to keep harmfulatoms from passing from the material of the crucibleinto the crystal.

I will not describe the apparatus used in such cases.Crystal growth was discussed in the second book of ourseries. As in the case of high-frequency currents, the laserbeam creates a small molten zone that slowly brings thematerial up to the growing crystal. I think it very prob­able that lasers will be supplanting other methods ofcrystal growth as well.

76Photons and Nuclei

metre yields a luminous intensity in the direction per­pendicular to the surface, equal 'to one candela.

The definition of luminous intensity and its unit aresurpr.isingly archaic. Specialists realize this and they hopethat m the near future a new basis will be chosen: energymeasurements and the standard luminosity curve. Thenluminous intensity can be defined by multiplying theset~o spectra! distributions. If we multiply these two func­tIOns and mtegrate the product, we obtain a measureof IUl~ino~s intensity. For the reader who is not acquaint­ed WIth mtegral calculus I can restate this as follows:mu~tiply !he i~ltensitiesof the separate spectralportions bytheIr lumm.osIty factors and then combine all the products.

At s~ffic18ntly.la~ge dista nces, a light source appearsas a pomt. That IS Just when it is convenient to measureluminous intensity. Now, around this point source letus c?nstruct a sphere and on the surface of the sphere aportIOn of area 8. Dividing 8 by the square of the dis­tance fr?m the centr~, we obtain the so-called solid angle.The UUlt of the solId angle is the steradian. If we cutout an area 8=1 square metre on a sphere of radius onemetre, the solid angle is equal to one steradian.

Luminous .flu:x is the luminous intensity of a pointsource multIplIed by the solid angle.

Do not be upset by the fact that the luminous flux be­comes. zero (vanishes) when we speak of parallel rays.That IS because the concept of luminous flux is not usedin such cases.

The unit of luminous flux is the lumen, which is equalto the flux delivered by a point source with a luminousintensity of one candela to an angle equal to one ste­ra~ian. The overall (total) luminous flux emitted by aPOInt in all directions is equal to 411: lumens.

Lu~ninous intensity characterizes a light source irre­spectIve of its surface. Yet it is quite obvious that theimpression will differ depending on the extent of the source.

2. Optical Instruments 77

For this reason, use is made of the notion of luminance(brightness) of a source. This is the luminous intensityper unit surface of the light source. Luminance is mea­sured in stilbs: one stilb is equal to one candela per squarecentimetre.

One and the same light source brings different luminousenergy to the page of an open book depending on wherethe source is located. What is important to the reader isthe illumination of the portion of his desk in which hisbook lies. Illumination is defined as the luminous intensi-

,ty divided by the square of the distance from the po~nt

source. Why the square? Because a luminous flux remaInSunchanged inside a given solid angle, no matter how farwe recede from the glowing point. Now, the area of thesphere and the area of the portion cut out by the solidangle will increase in inverse proportion to the squareof the distance. This simple rule is called the inverse­square law. If the distance of the book from a small lampis increased from 1 metre to 10 metres, we thus reducethe illumination of the page by a factor of one hundred.

The lux is the unit of illumination. This is the illumi­nation generated by a luminous flux equal to one lumenover an area of one square metre.

The illumination on a moonless night is equal to 0.0003lux. On a night with full moon the illumination canreach 0.2 lux. For comfortable reading we need an illu­mination of 30 lux. When shooting movie scenes, powerfulprojectors are used that raise the illumination to 10 000lux.

So far we have not mentioned the instruments thatmeasure luminous fluxes and illumination. At the presenttime, such measurements are no problem. Actually, weproceed exactly as we did in providing the new definitionof the candela. We measure the energy falling on a pho­tocell and graduate the scale of the photocell in unitsof lux with account taken of the luminosity curve.

Figure 2.9

Take a look at Figure 2.10. The object is sh~wn a~ov~,the "image" below. Yes, this is the image. It IS an llltn­cate combination of dark and light rings called a hologramand is indeed the image of the object, only it is a latent(hidden) image. The hologram contains complete infor­mation about the object; it would be more correct tosay complete information about the electromagnetic wavescattered by the chess pieces. A photograph does notcontain so much information. The best photograph con­veys precisely all information about the intensi~y of thescattered rays. But a wave scattered by any. po~nt of .anobject is fully characterized not only by ItS ~ntens~ty

(amplitude) but also by its p~Iase. A hologr~m IS an. lll­terference pattern and each lIght or dark hne deSCrIbes

79

FROMLASER

-------------------~~ IMAGE

-------

2. Optical Instruments

Holography

The invention of lasers has opened up a new chapterin the development of science and technology. It is hardto find a branch of knowledge in which stimulated emis­sion has not revealed fresh vistas.

In 1947, Dennis Gabor (1900- ) outlined the principlesof obtaining images of objects in a radically different wayfrom photography. The new technique is called holo­graphy. Holography became possible because of the pe­culiarities of stimulated emission that make it differentfrom ordinary light. Let us stress once again that inlaser radiation nearly all photons coincide in all theirfeatures: frequency, phase, polarization, and directionof propagation. A laser beam is hardly at all spread out,which means an extremely narrow beam can be thrownto great distances from the source; laser beams havethe property of a very great coherence length. It is dueto this circumstance (which is particularly importantin holography) that interference of split rays with a largepath difference is possible.

The upper part of Figure 2.9 illustrates the techniqueof obtaining a hologram. The object of interest is illum­inated with a broad and weak (so as not to harm theobject) laser beam. The same beam is scattered by theobject and by a mirror, which generates a so-called refer­ence wave. The two waves are superimposed, interferenceoccurs, and the pattern is recorded on a photographicplate.

Photons and Nuclei 78

Photometers in use last century operated on the prin­ciple of comparing the luminances (brightnesses) of twoadjoining illuminated areas. One area received light thatwas to be measured; on the other, using simple devices,one reduced the luminous flux a known number of timesuntil the two adjoining areas had the same illumination.

Photons and Nuclei

Figure 2.10

80 2. Optical Instruments 81

and phases as those that created the hologram originally.These waves combine to form a wave front that is identi­cal to the wave front that produced the hologram. Whatoccurs is a peculiar reconstruction of the wave when thehologram is illuminated in the same conditions that theobject was illuminated. The result is an image of theobject.

Research in the field of holography continues. We arenow able to obtain coloured images. Improved results arepossible by taking several holograms from different an­

,gles. Finally (and this is perhaps the most importantthing) it turns out that we can examine holograms with­out resorting to lasers.

There are books which treat the subject of holographyin detail. Holography merits our attention for the reasonthat it is high-capacity method for storing three-dimen­sional information concerning an object. Future researchwill undoubtedly bring holography into new fields oftechnology and into the home.

not only the intensity but also the phase of the rayscoming from the object onto the appropriate spots of thephotographic plate.

A hologram, like any ordinary photographic plate, canbe developed, fixed, and stored for any length of time.Whenever we want to, we can view the object by irra­diating the hologram, as shown in the lower part ofFigure 2.9, with light from the same laser, thus reconstruct­ing the original geometric arrangement: the laser beamis directed like the beam reflected from mirror. Then animage of the object appears where the object once stood,and that image, ideally, is identical to the picture thatconfronted the eye.

We will not go into the theory of obtaining holograms.The basic idea is that when a hologram is illuminated,there arise scattered waves with the same amplitudes

3. Hard ElectromagneticRadiation

The Discovery of X Rays

X rays is the term used for radiation occupying aportion of the electromagnetic spectrum approximatelyfrom several tens of nanometres to 0.01 nanometre. Thestill harder (that is, having still shorter wavelengths)rays are called gamma rays.

As we have already mentioned, the names for thevarious portions of the electromagnetic spectrum are rath­er conventional. The term designating each portion ofthe spectrum has less to do with the wavelength thanwith the nature of the source of radiation. Most frequent­ly, the term "X rays" is used for radiation that occurswhen a flux of electrons encounters an obstacle.

Wilhelm Conrad Roentgen (1845-1923) discovered thiskind of radiation on November 8, 1895. During thoseyears, many physicists around the world were studyingfluxes of electrons arising in evacuated glass tubes (illus­trations of these are given in Figure 2.6 of the third bookof this series). Two electrodes were soldered into a vesseland a high voltage applied. The fact that some kind ofrays were emitted from the cathode of such a tube hadbeen suspected for quite some time. At the very beginningof the 19th century a number of researchers observedflashes inside the tube, a fluorescence of the glass.Through the experiments of the German physicist JohannWilhelm Hittorf (1824-1914) and the English chemistand physicist William Crookes (1832-1919) it was demon-

3. Hard Electromagnetic Radiation 83

strated that these were rays, in the path of which anobstacle had been placed. All textbooks included a pho­tograph of the Crookes tube with a cross which he madein 1878, nine years after Hittorf. The cross threw adefinite shadow onto the glass. This elegant experimentproves that some kind of rays emanate from the cathodeand move in straight lines. When the rays fall on theglass, it glows and a thin layer of metal absorbs theradiation.

The fact that cathode rays are a flux of electrons wasproved by Sir Joseph John Thomson (1856-1940) in 1897.By a method discussed in the third book of this serieshe was able to determine the ratio of the charge to themass of the electron. Another 10 to 15 years passed andit became clear that the electron is a minute particleof electricity.

But we are going astray. What interests us now is thediscovery made by Roentgen. However, our brief digres­sion was made to emphasize the fact that Roentgen'sdiscovery preceded an understanding of the nature of therays emitted hy the cathode. Actually, it was preciselybecause of a lack of understanding that Roentgen wasinvestigating a variety of tubes with different locationsof electrodes and different forms of glass envelope.

The events of the evening of November 8, 1895 areperfectly well known. Roentgen placed a piece of blackcloth over the tube, turned off the light in the room,and was about to go home with the light switch left on.Then he glanced at the instrument he was working withand noticed that the screen covered with barium plati­nocyanide (which is capable of fluorescing) next to thetube was glowing. Roentgen returned, threw the switch,and the fluorescence ceased. He turned on the switchand the screen began to glow anew. Roentgen knew thatin the tube of the design he was working with, cathoderays could not pass through the protective covering thrown6·

over the tube and then through a big layer of air as well.That could only mean one thing-a new hitherto unknownkind of radiation.

Roentgen first reported his discovery at the end ofthe year. During this interval he had made such athorough study of the properties of the new rays that upuntil the discovery of the diffraction of X rays by a crys­tal (1912)-we will come to that soon-nothing new aboutthe X rays was unearthed. The name X rays was givenby Roentgen himself, but in some countries the radia­tion is termed Roentgen rays.

The most remarkable property of X rays is the propertythat Roentgen studied and illustrated from the very start:their ability to pass through materials that are opaqueto light. (Figure 3.1 is a reminder of the many caricaturesthat appeared two or three months after Roentgen's firstpublication.)

The penetrating power of X rays was a boon to medi­cine. It also became possible to detect imperfections inindustrial goods. The amazing results of roentgenographystem from the fact that materials of different densityabsorb X rays to different degrees. The lighter the atomsof the material the less fhe rays are absorbed.

Photons and Nuclei

Figure 3.t

84 3. Hard Electromagnetic Radiation 85

It was also soon found that the penetrating power ofthe rays increased with increasing voltage in the tube.Voltages ordinarily used in X-ray equipment are betweenseveral tens and several hundreds of kilovolts.

Research into the properties of X rays elicited thefact that they are caused by retardation of electron fluxesdue to an obstacle. It is curious thing to recall that forquite a long time Roentgen tubes were manufactured withthree electrodes. An "anticathode" was fitted oppositethe cathode toreceive the impinging electrons. The anodewas placed to one side. A few years later it became clearthat that was a needless complication and today twoelectrodes are used. A flux of electrons impinges on theanode, the surface of which is beveled. Then the fluxof X rays is directed where needed. If the surface of theanode meets the electron flux at right angles, the rays gooff the anode in all directions and we lose a certainamount of intensity.

The use of X rays created a revolution in industryand medicine. X-ray technology today is a highly refinedfield. By varying the position of an object relative tothe X-ray tube, we can obtain several pictures, thusenabling us to establish the exact position of a defectnot only in projection but also its depth within the ma­terial.

Depending on the materials or tissues being investigat­ed, use is made of hard (more penetrating) radiation orsoft radiation. The main purpose is to attain a contrastso as to see the defect, which may differ in density onlyslightly from the main material.

The law oj absorption of X rays, like the law of absorp­tion of any other radiation, is quite obvious. What in­terests us is how the intensity of radiation (recall thatintensity is energy per unit time and unit area) changesafter passing through a plate of thickness d. Since thisbook is for the reader who is not acquainted with integral

Photons and Huclei 86

calculus, I will confine myself to the statement of tholaw for radiation passing through thin plates. "Thin"stands for the intensity falling off very slightly, sayabout 1%. In that case the law is simple: the portionof absorbed radiation is directly proportional to the thick­ness of the plate. If the intensity diminishes from a value10 to a value I, then this simple rule can be written thus:

1 -/0-1-= p,d

The proportionality factor p, is called the absorptioncoefficient.

Here is a simple question I have posed at examinations:In what units is the absorption coefficient measured?This is an easy question. The units of measurement haveto be the same on both sides of the equation. That'sclear. You can't say that 10 kilograms is more than 5 me­tres. The only comparisons can be between kilograms andkilogra~s, amperes and amperes, ergs and ergs. And sothe Ulllt~ are the same on both sides of the equation.

Now, 1Il the left-hand member of this equation we havea dimensionless number. By saying that the portion ofabsorbed radiation is equal to 1130 or 0.08, we have saidall there is to say. The units of measlirement cancel outwhen intensity is divided by intensity. And that meansthat the quantity on the right must be nondimensional.Since thicknoss is measured in centimetres (or other unitsof length), the absorption coefficient is measured in in­verse centimetres, that is, cm- I •

Suppose a ray passes through a lO-cm-thick plate andloses 1 % of its intensity. The left-hand side of the equa­tion is equal to 1/100. This means the absorption coeffic­ient here is equal to 0.001 cm- I . Now if the radiationis soft and loses one percent of its energy when passingthrough a foil one micrometre (0.0001 cm) in thickness,then the absorption coefficient will be 100 cm-I •

3. Hard Electromagnetic Radiation 87

Physicists do not have a good theory for establishinga formula for the absorption coefficient. I can only saythat the absorption coefficient is roughly proportionalto the cube of the wavelength of X-radiation and to thecube of the atomic number of the substance throughwhich the radiation is passing.

Since the wavelengths of X rays are extremely short,the frequencies of the electromagnetic waves are high.This means that a quantum of X rays, hv, carries atremendous energy. This energy is only sufficient forchemical reactions leading to blackening of the emulsionof a photographic plate and the phosphorescence of screens(even light waves can do that), but there is even morethan enough to break up molecules. In other words,X rays ionize air and other media through which theypass.

Now a few words about gamma rays. This term appliesto the short-wave radiation that originates in radioactivedecay. Gamma rays are emitted by naturally radioactivesubstances and are produced by artificial elements. Anuclear reactor generates gamma radiation of course. Veryhard (high-energy) gamma rays are produced in the ex­plosion of an atomic bomb.

Since gamma rays can have a very short wavelength,their absorption coefficient can be very small. For instance,the gamma rays that are emitted in the disintegrationof radioactive cobalt are capable of passing through tensof centimetres of steel.

Substantial doses of short-wave electromagnetic radia­tion (such that is capable of disrupting molecules) arevery dangerous to any living organism. Therefore protec­tion is needed when dealing with X rays and gamma rays.Ordinarily it is in the form of lead. The walls of X-rayrooms are covered with a special coating contaillinO'barium salts. '"

Gamma rays, like X rays, can be used for radioscopy.

Photons and Nuclei RS

Usually use is made of the gamma rays of radioactivesubstances which are the "ashes" of nuclear fuel. Theiradvantage over X rays is the extreme penetrating power,but most important of all is the possibility of using,as the source of radiation, a small ampoule that can befitted into places that are inaccessible to an X-ray tube.

X-ray Diffraction Analysis

In 1912 Roentgen was head of the chair of physicsat the University of Munich. Problems dealing with thenature of X rays were constantly being discussed. I mustsay that Roentgen, who was an experimental physicisthimself, had much respect for theory. There were a goodmany talented theoretical physicists at the chair of physicsat the University of Munich that were cudgellinO' theirbrains over the nature of these new X rays. <>

And of course attempts were made to investigate thenature of the X rays by passing them through a diffrac­tion grating. Using a diffraction grating, one can provequite definitely the wave nature of light and also obtainexact determinations of the wavelength of one or anothertype of radiation.

One way of making such a grating is to rule finescratches on a glass plate coated with aluminium; this isdone with a cutting tool made of ivory. The lines thus cutmust be strictly equally spaced. A good grating musthave a small period and a large number of lines. Up tohundreds of thousands have been ruled and over a thou­sand per millimetre.

With the help of a lens, a strong point source of lightcan produce a parallel beam of light that falls on thegrating at right angles. The rays emerge from each aper­ture and proceed in all directions (in other words, eachaperture becomes the source of a spherical wave). Butonly in certain directions will the waves from all aper-

3. Hard Electromagnetic Radiation S9

tures be in-phase. For reinforcement, it is necessary thatthe path difference be equal to an integral number ofwavelengths. Strong rays will go in directions at anangle a that obey the condition

a sin a=n').

where n is an integer and a is the distance between slits.The reader can easily derive this formula without our help.

The whole number n is termed the order of the spec­trum. If monochromatic light is incident on the grating,we obtain, in the focal plane of the eyepiece, severallines separated by dark intervals. If the light consists ofwaves of different wavelength, the grating will produceseveral spectra: of first, second, etc. orders. Each suc­ceeding spectrum will be more extended than the pre­ceding spectrum.

Since the wavelength of the light is of the same orderas the distance between the slits, diffraction gratingsdisperse light (and not only visible light but also ultra­violet radiation and, best of all, infrared radiation) intospectra. These instruments make it possible to carry outa detailed spectral analysis of the radiation.

But in relation to X rays, diffraction gratings behavedlike a system of open doors. X rays passed through themwithout deviating. It could be suspected that X rays area flux of particles. But, on the other hand, there wasevery reason to think that this X-radiation is the sameelectromagnetic field as light, only with the wavelength ').much shorter. True enough, suppose the wavelength isvery short. If that is so, then, according to the diffractionequation, all n rays proceeding at diffraction angles afrom the line optical grating a sin a=n'). will practicallymerge and no diffraction will be apparent. But to makea diffraction grating with slits separated by a distancea equal to millionths of a micrometre is out of the ques­tion. So what have we?

Photons and Nuclei 90

A young German physicist Max von Laue (1879-1960)was sure that X rays are a type of electromagnetic radia­tion. His acquaintance was a crystallographer who wasconvinced that a crystal is a three-dimensional latticeof atoms. In one of their numerous talks on scientifictopics, Laue got the idea of correlating his view of thenature of X rays with the concept of a crystal as a lattice.Laue's idea was this: suppose the distances between theatoms of a crystal and the wavelength of X rays werequantities of the same order.

Can a three-dimensional lattice take the place of aline grating of slits? It was not obvious at all, but Lauedecided to try it. The first experiment was simplicityitself. A beam of X rays was diaphragmed. A large crystalwas placed in the path of the rays, and next to the crystala photographic plate. True, it was not at all clear whereto put the plate since the crystal was not a line lattice.The first attempts were thus failures. But soon, due to amistake, they got the plate in the proper position.

This accidental mistake did not play any special rolein the discovery because Laue was also working on thetheory of the phenomenon besides trying to detect it.He was soon able to extend the theory of the line dif­fraction grating to the three-dimensional case. From thetheory it followed that diffracted rays would appear onlyin the case of certain orientations of the crystal withrespect to the incident rays. From the theory it also fol­lowed that the most intensive rays were those deflectedat small angles. And from this it was evident that thephotographic plate had to be positioned behind thecrystal perpendicular to the incident rays.

Among the first researchers to notice this discoverywere the Braggs, the father Sir William Henry (1862-1942)and his son Sir William Lawrence (1890- ), English phys­icists. They repeated Laue's experiment, provided a verysimple and pictorial interpretation and also demonstrated

3. Hard Electromagnetic Radiation 91

many simple instances in which Laue's discovery could beused as a method for studying the atomic structure ofsubstances.

Let us take a brief look at X-ray diffraction analysisand also examine the path by which we can determinethe structure of a crystal, that is, measure the distancesbetween atoms with an accuracy of up to one hundredthof an angstrom and give a picture of the spatial arrange­ment of atoms in a molecule, and also determine thenature of the packing of molecules in a crystal.

Suppose a crystal is set in a special holder and rotateson an axis. The X ray falls at right angles to the axis ofrotation. What happens? Let us examine the diffractionphenomena that take place when an X ray is incident onthe crystal so that a lattice point is regarded as thescattering centre.

The Braggs (father and son) demonstrated that thescattering of X rays by lattice points is equivalent to acurious kind of selective (that is, taking place only forcertain discrete values of the angle) reflection of raysfrom a system of nodal planes, into which the lattice maybe partitioned.

Suppose a ray, which is an electromagnetic wave of adefinite length, falls on a crystal at some angle. Thisangle wiII differ for different systems of planes. We haveevery right to assume that any atomic plane wiII reflectthe X ray in accord with the law: the angle of incidenceequals the angle of reflection. But there is essential dif­ference from optical rays. Unlike light, X-radiation pen­etrates deep into a crystal. This means that the reflec­tion of the ray wiII occur not from the external surfacebut from all atomic planes.

Let us consider one such system characterized by theinterplanar spacing d. Each of them will "reflect" anincident ray at one and the same angle e. These reflectedrays must interfere, and a strong secondary ray will arise

onl~ if the rays reflected from all planes of the familyare In the same phase. In other words, the path differencebetween the rays must be equal to a whole number ofwavelengths.. Figure 3.2 depicts a geometric construction from whichIt follows that the path difference between adjacentr~flecte~ rays is equal to 2d sin e. Hence the condition ofdIffractIOn will be of the form

2d sin e=n'A.

The Russian crystallographer G. V Vulf arrived at thisformula at the same time as the Braggs and so it iscalled the Bragg-Vult equation (also known as Bragg'slaw or Bragg's equation).. A crystal can be partitioned into systems of planesIn nU~berless ways. But only a certain system becomeseffec~Ive for reflection: that in which the interplanarspacIng and orientation with respect to the incident rayare such that Bragg's equation is obeyed.

Obviously, if the ray is monochromatic (that is, thee!ectromagnetic wave has a definite length), then reflec­tIOn may not OCcur in the case of an arbitrary positionof .the crystal with respect to the ray. However, by 1.'0­

!atIng the crystal we can bring different systems of planesmto the reflecting position in turn. This appeared to bethe most suitable way for practical work.

Photons and Nuclei

Figure 3.2d sin 6

92 3. Hard Electromagnetic Radiation 93

As for Laue's experiment, its success stemmed from thefact that the radiation incident on the crystal was the"white spectrum" of X rays, that is, a flux of waveswhose lengths are continuously distributed over a certaininterval (see below). Therefore, although the crystal wasfixed in Laue's experiment, different systems of planesappeared in the "reflecting" position for waves of differentlength.

At the present time, X-ray diffraction analysis is com­pletely automatic. A small crystal (0.1 to 1 mm) is fixedin a special holder, which can turn the crystal accordingto a specified programme, bringing into the reflectingposition all its system of planes one after the other.Every reflecting plane (we'll use that word so as not torepeat "system" all the time) is characterized, first, byits interplanar spacing, second, by the angles which itforms with the axes of an elementary cell of the crystal(lengths of the edges and the angles between the edgesof the cell are measured first of all and automatically)and, third, by the intensity of the reflected ray.

The more atoms a molecule contains the greater (na­turally) the dimensions of the elementary cell. This in­creased complexity brings with it a greater volume ofinformation. The point is that the larger the cell thegreater the number of reflecting planes. The number ofmeasured reflections can range from several tens to sev­eral thousands.

We promised a brief survey of the idea underlying X-raydiffraction analysis. Let us first turn the problem around.Suppose we know everything about the structure of acrystal. This means we know the pattern of atoms, thatis, we know the coordinates of all atoms that make upan elementary cell (I suggest the reader go back to thesecond book of this series and refresh his memory con­cerning crystal structure). Let us consider some kind ofsystem of reflecting planes. Obviously the following is

Photons and Nuclei 94

sufficient. If most of the atoms of the crystal lie in planespassing through the lattice points, then all atoms willscatter the X rays in a single phase. What we get is astrong reflected ray. Now take another case. Half of theatoms lie in the nodal planes, and the other half liein between the reflecting planes. Then half of the atomsscatter incident light in one phase while the other halfscatter it in the opposite phase. There will be no reflec­tion!

These are two extreme cases. In all other cases weobtain rays of different intensity. A measuring instrument(called an automatic diffractometer) is capable of measur­ing the intensities of reflections that differ by a factor often thousand.

The intensity of a ray is uniquely connected with thearrangement of the atoms between the nodal planes. Theformula stating this relationship is too unwieldy to giveand, what is more, we don't even need it. What we havealready described about the two extreme cases shouldbe enough for the reader to be convinced there is a for­mula in which intensity is represented as a function of thecoordinates of all atoms. The type of atoms is also takeninto account in this formula because the more electronsthere are in an atom the more intensely it scatters X rays.

Also included in the formula relating structure andintensity of reflected rays is of course information aboutthe orientation of the reflecting plane and also about thedimensions of the elementary cell. We can write down asmany such equations as there are measured reflections.

If the structure is known, then the intensities of allrays can be computed and correlated with experiment.But that isn't the problem that confronts us! What wehave before us is the converse problem: using informationconcerning the intensities of several tens or hundreds orthousands of reflections, find the coordinates of all atoms ina cell. At first glance it might seem that with the tremen-

3. Hard Electromagnetic Radiation 95

dous calculating power of modern computers the converseproblem can be solved with ease. A lot of equations? Butcomputers can surely handle them.

However, the matter is not at all so simple. Experimen­tal findings have to do with the intensities of the rays.The intensity is proportional to the square of the ampli­tude. The formula discussed above is actually a formulaof interference. The waves scattered by all atoms of acrystal interfere with one another. What we have is acomposition of amplitudes scattered by all atoms. Theoverall amplitude is computed and the intensity is foundby squaring the amplitude. To solve that problem is easy.But how do we handle the converse problem? Take thesquare root of the intensity in order to obtain the am­plitude? Correct. But a square root has two signs.

I just hope the complexity of the problem is now clearto the reader. We have equations enough (more thanenough) to find the coordinates of the atoms. But in theright-hand member of the equation we have numbers thatare known only up to sign.

We are thus at an impasse. And, at first, researchersdid not attempt to solve the converse problem. Theyemployed the trial and error method. They assumed,on the basis of information concerning allied structures,that the unknown structure was such and such. Thenthey calculated the intensities of a dozen rays and com­pared them with experimental findings. And failed. Andthen they took up another model of structure.

In simple cases this approach produced correct results,though there were difficulties. But when the "structuremen" (which is what these researchers were being called)had studied nearly all the simple structures, they came

.to a halt when confronted with the converse problem.In the middle of the 1930s it was conjectured that even

complex structures might be "solved" (that was the lingothen) if one con fined himself to a study of molecules

vented and is reduced to the construction of what areknown as Fourier series of electron density. Only a readerwith a good mathematical background could stomachany discussion of the theory of Fourier series or, stillworse, its application to the problem of determiningstructure. But we won't need such knowledge here. Themain thing is that the idea as such has been discussed.I think that is sufficient for our purpose.

In what form does the physicist (the specialist in X-raydiffraction analysis) provide the chemist with informa­tion concerning structure? Some idea of this can be gainedby a glance at Figure 3.3, which shows a very simplestructure called ammonium barbiturate. Today, determin­ing the structure of such complexity is child's play. An7-2542

Photons and Nuclei 96

that contain many light atoms and one heavy atom. Aheavy atom contains many electrons and scatters X raysmuch more strongly than light atoms. Therefore, to afirst approximation (very rough) it may be assumed thatthe crystal consists solely of heavy atoms. If there isone atom in a cell, its coordinates can be found with rela­tive ease by means of trial and error. With these coordi­nates, it was then assumed that only that atom functionedin the crystal, and it was further assumed that thesigns of the amplitudes determined for the fictitious struc­ture consisting of only heavy atoms were the same asfor the actual structure.

A highly important discovery was made some twentyyears ago: namely, proof was given of a theorem concern­ing the relationship between the amplitudes of reflectionsof different families of planes. For example, a relationshipwas found between the signs of the amplitudes of threereflections that are shifted in phase with respect to thepoint of the cell by a, /3, and a+/3. It turns out that ifthe product cos a cos /3 cos (a+/3) exceeds 1/8 in absolutevalue, then it must have a positive sign. You can checkfor yourself if you don't believe me.

The development of that idea led to what are known asdirect methods of structure analysis. Even in rather com­plicated cases the experimental device can be hooked upto a computer, which will then "spew out" the structureof the crystal.

When the signs of the amplitudes of reflection havebeen established, then determining the coordinates of theatoms becomes, as we now know, a problem involving thesolution of a very large number of equations with manyunknowns. It is important here that the number of equa­tions be at least ten (better, a hundred) times the numberof desired coordinates of the atoms.

I will not go into the techniques used to solve suchsystems of equations. Actually, the problem is circum-

3. Hard Electromagnetic Radiation

Figure 3.3

97

with small openings) and make the beam impinge on acrystal, then in the most general case we obtain severalrays reflected from planes that are in positions that sat­isfy the Bragg-Vulf equation. If we position the crystalso that some plane (yielding a strong reflection) coincideswith the axis of rotation of a special instrument (calledan X-ray spectrograph) and then turn the crystal so thatthe plane comes under the incident ray at a succession ofall angles e, then a component of the spectrum witha definite wavelength will be reflected for each positionof the crystal. This reflected wave can be either pickedup by an ionization counter or recorded on film. In thisway we can create a monochromatic ray of any wavelengthof the radiation spectrum and, also, develop a methodfor exploring the spectrum of any type of radiation.

A typical spectrum of an X-ray tube with molybdenumano~e is shown in Figure 3.4 (at 35 kilovolts). The con­clUSIOn is almost immediate that there are two reasonsfor the generation of an X-ray spectrum. We see that theobserved spectrum is a superposition of sharp peaks on,.

Photons and NucleI 98

automatic device gives the structure without any efforton. the part of the researcher. The electronic computer canprmt out the numbers (the values of the coordinatesof the atoms) or can produce displays like that of Fiaure3.3. Atoms of different kinds are symbolized by ci;clesof di.fferent ~ize. And if the researcher wants the computerto gIve a pIcture of electron density, it can do so. Eacha.tom is displayed. in th.e way that geographers portrayhnes of equal altItude m mountainous regions. But inour case the closed circuits are not lines of equal altitudebut cu.rves indicating the electron density at every point.The tIp of the "mountain peak" is the centre of theatom.

The accompanying diagram is a very minute portionof the contribution that this method has made to science.The success of the method is colossal. To date, the struc­tures of over 15 thousand crystals have dbeen decipheredand these include several tens of structures of protein~whose molecules consist of many thousands of atoms.

. Determining the structures of complex molecules laysthe ~oundation of b~ological chemistry and biologicalphYSICS, two new SCIences that have burst into bloomand are expected to give us discoveries of the secrets oflife, illness, and death. Despite its sixty-some years ofexistence, X-ray diffraction analysis remains at the fore­front of science.

The X-ray Spedrum

In the preceding section we mentioned the existence ofa "white" spectrum and monochromatic rays. How canwe decipher the nature of the spectrum of hard electro­magnetic radiation? And when is it "white" and in whatcases is it monochromatic?

If we diaphragm X rays or gamma rays coming fromsome source (that is, if we place in their path two screens

3. Hard Electromagnetic Radiation

Figure 3.4

ItS .!::1t t

I-

7\~

i'1

0.5 0.7 0.9 0

A,A

99

Photons and Nuclei 100

a continuous curve. Of course, the origin of these peaksdiffers from the origin of the solid curve.

Immediately after the discovery of the diffraction ofX rays, a study began of X-ray spectra and the followingwas established. A continuous spectrum is not typical ofthe material of the anode and depends on the voltage.Its peculiarity is that it breaks off sharply at a certainminimum wavelength. When moving in the direction oflong wavelengths, the curve passes its maximum andsmoothly falls off without having any apparent endpoint.

By increasing the voltage in the X-ray tube, researchersdemonstrated that the intensity of the continuous spec­trum grows and the boundary is shifted towards the shortwaves. And the following very simple equation was estab­lished for the boundary wavelength:

'A . _ 12.34mm--U-

In quantum language, the statement of the rule is simple.The quantity eU is the energy gained by an electron incovering the distance from the cathode to the anode.Naturally, the electron cannot release more energy thanthis quantity indicates. If it transfers all the energy togenerating an X-ray quantum (eU =hv), then after sub­stituting the constants we obtain the above equation('A is in angstroms and U in kilovolts).

Since we get a continuous spectrum, it follows that theelectrons do not necessarily give up all their energy to thegeneration of X rays. Experiment shows that a largepart of the energy of an electron beam is converted intoheat. The efficiency of an X-ray tube is very low. Theanode heats up and it has to be cooled by means of waterflowing inside the anode.

Is there a theory that can explain the reason for acontinuous spectrum of X rays? Yes, there is. Calculations(which unfortunately we cannot give here) show that

3. Hard Electromagnetic Radiation 101

from the general laws of the electromagnetic field (fromMaxwell's equations) that were discussed in book threeof this series, it follows very rigorously that if electronsare decelerated, the result is a continuous spectrum ofX rays. Collisions with a solid are not essential at all.If the electrons are decelerated by means of a counter­field, the result is a continuous emission of X rays withoutany participation of the anode in the game.

There is yet another way of encountering a continuousX-ray spectrum. We recall that a continuous electro­magnetic spectrum is emitted by incandescent bodies.In terrestrial conditions we do not encounter an X-rayspectrum of such origin because (compare the formulagiven on page 15) at the highest possible temperatureof an incandescent body (6000 degrees-since no solidbody can stand up to a higher temperature) the wavelengthof thermal radiation will be close to half a micrometre.But we must not forget about the existence of plasma.In stellar interiors and in artificial plasma produced hereon earth, temperatures in the millions of degrees can beobtained. Then the thermal spectrum of electromagneticradiation will also include X rays. X rays coming from out­er space help resolve intriguing problems of astrophysics.

Now let us touch on the sharp peaks that are superim­posed on the curve of the continuous spectrum.

Just the reverse was proved with respect to these rays,the reverse with respect to the law of the continuousspectrum. The sites of the peaks, that is, their wave­lengths, are unambiguously determined by the materialof the anode. Such emission therefore goes by the namecharacteristic.

Its origin is unbiasedly accounted for by the quantummodel of the atom. The electron rays of an X-ray tube arecapable of penetrating an atom of the anode and knockingout electrons from the very lowest energy levels. As soonas a low level is vacated, that site is filled by an electron

Photons and Nuclei 102

mo~e dis~ant from the centre of the atom. Energy isemItted In accordance with the basic quantum law:J!m-.En=hv. Energy levels are arranged in different waysIn dIffere~t atoms. It is therefore natural that the newspe~tra wIll ~e characteristic spectra.

SInce the hnes of a characteristic spectrum are thes~ronge~t ones, the¥ a.re used for X-ray diffraction analy­SIS. It IS best to ehmInate the continuous spectrum' th ti~, b~fore .making the ray fall on the crystal under inve~­tIgatlOn, It has to be reflected from the crystal monochro­mator.. Since. the spectra of different elements are characteris­

tic, a dIspersIOn of t~e ray into a spectrum may be usedfor purposes of chemICal analysis. It is called X-ray spec­tral analysis. There is a whole range of fields (for examplethe study of rare-earth elements) where X-ray spectralanalysis is absolutely indispensable. The intensities ofspectral X-ray characteristic lines make it possible todetermine to a high degree of accuracy the percentagecontent of any element in a mixture.

Now a word ?r two ~o.ncerning the spectra of gammarays. In terr.es.tnal ~ondIt~ons, we have to do with gammarays that ongInate In radIOactive decay (this will be dis­cllssed later on) .. Radioactive disintegration mayor maynot be accompamed by the emission of gamma radiation.But no matter what the type of radioactive decay thespectrum of ~a~ma radiation will be characteristic.

If cha.ractenstIC X rays appear when an atom fallsfrom a hIgh energy level to a lower level, then gamma raysare p.roduced as a result of a similar transition of theatomIC nucleus.

The gamma-ray spectra of radioactive transformationshave been thoroughly studied. There are tables in whichone can find e~act data on the wavelengths of gammarays generated In alpha- and beta-transformations of oneor another radioactive isotope.

3. Hard Electromagnetic Radiation

Radiography of Materials

I have time and again had to call attention to thefact that terminology is a weak spot in science. Sciencedevelops at such a rate that the meaning of a term changesin the course of a single generation. And at the sametime, changes in terminology are associated with altera­tions of what is taken to be accepted. What is more, oldbooks cannot be taken out of circulation, and so the onlything left is to include strict definitions of every termthat is employed and to redefine others.

At the present time, when one speaks of X-ray diffrac­tion analysis, he has in view the investigation of theatomic structure of crystals. The object of study is asingle crystal of the substance.

However, studying structure by means of X rays doesnot in any way exhaust the problem. Characteristic andinformation-rich pictures are obtained if radiographs aretaken of any materials and not only single crystals.In such cases we ordinarily use the term radiography.

If a piece of metal foil is placed in the path of a mono­chromatic X-ray beam, a system of concentric circles ap­pears on a flat photographic plate. A radiograph of thistype is termed a Debye crystallogram. What is its origin?

Most solids consist of tiny crystals oriented at randomwith respect to one another. When a melt of some sub­stance begins to cool, crystallization sets in simultaneous­ly at a large number of sites. Each minute crystal growsevery which way and the growth continues until thecrystals meet.

Each crystal has the same system of atomic planes;this is because we are dealing with structurally identicalcrystals. Let us examine one of these systems of planeswith interplanar spacing d. There are a large numberof minute crystals (ordinarily, the linear dimensions ofa solid-body crystal is, as to order of magnitude, equal

to one ten-thousandth of a centimetre), and among themare those whose planes lie at an angle e to the incidentray, which angle satisfies the Bragg-Vulf equation. Eachsuch crystal will leave a spot on a photographic plate.Reflections will be produced by all such minute crystalswhose ~ormals to the planes form a cone (Figure 3.5).If that IS so, then the reflected rays will lie in the cone.The intersection of that cone with the photographic plateyields a circle. By measuring the radii of the circlesand knowing the distance from the object to the photo­graphic plate, we can find the Bragg angle e at once andwill then be able to calculate all the interplanar spacingsof th.e substance on the basis of such a radiograph.

Usmg such a diffraction pattern, we can at once distin-

Photons and Nuclei

(I)-.J

tIl:

PRIMARY ~

_B_EA_M__ ~ _u _ --h'-1E~W.,...1

~ 90~ ()ou

Figure 3.5

104 3. Hard Electromagnetic Radiation 105

guish any amorphous material from crystalline material.Amorphous bodies do not have reflecting planes. There­fore the radiograph will not reveal any system of distinctdiffraction rings. There is always some sort of order inthe arrangement of molecules of a substance for the sim­ple reason that atoms cannot "overlap" in any way. Andas calculations show, this results in the radiograph of anamorphous body having one or (rarely) two smeared rings.

However, the rings observed on radiographs yield agood deal of valuable information concerning the struc­ture of materials such as metals, polymers, and naturalcompounds. If the substance consists of large crystallites,then the diffraction ring will not be continuous and willconsist of separate small spots. If the crystallites are notarranged haphazardly but are oriented along an axis orplane, as occurs in wire or sheets of metal, in polymerstrands and in vegetable fibres, then the diffraction ringswill tell us so immediately. It is easy to see that if wehave preferential orientations of crystallites, the reflec­tions from atomic planes will not fill the cone of rays ina continuous fashion. In place of rings, the radiographreveals arcs. And if the orientation is highly refined,these arcs may degenerate into tiny spots.

Of course, to give a detailed description of the typeof structure from the form of radiograph is no easy prob­lem. In this case, too, the method of trial and errorplays an essential role. The investigator thinks up modelsof the structure of materials, calculates patterns of reflec­tions of X rays that should yield the models he has Con­cocted and, by comparing calculations and experiment,chooses the proper pattern for the structure of the sub­stance.

In the radiography of materials we distinguish, some­what artificially, between scattering through large andsmall angles. From the Bragg-Vulf equation given above,it is clear that scattering through large angles occurs

Photons and Nuclei 106

if a structural periodicity is observed at small distances(say, 3 to 10 A). If the reflected (we could say, scattered)X rays yield a diffraction pattern that collects near theprimary beam, then this means the structure possessesperiodicity at large distances.

In the science of metals we deal mainly with diffrac­tion rings located at large angles, since they consist ofcrystallites whose atoms form regular lattices with cellshaving dimensions of the order of units of angstroms.

When the objects of investigation are substances madeup of. macromolecules, we encounter a very interestingsItuatIOn. (By the way, macromolecules are found inmany natural substances such as cellulose or DNA andalso synthetic polymer materials with such familiarnames as polyethylene, nylon, capron and so on.) In somec~ses we obtain radiographs exhibiting rings of largedIameter only. In other words, we have to do with thesame kind of scattering through large angles as is thecase in metals. And at other times, on occasion, we donot find large-diameter rings but rather diffraction raysthat only slightly deviate from the primary direction.Finally, there are cases where the substance reveals X-rayscattering both through large and small angles.

Small-angle scattering (of course the division into typesof small-angle and large-angle scattering is somewhatartificial) is in the range from several minutes to 3-4°.Naturally, the smaller the diffraction angle the greaterthe repeating period of structural elements that producedthe diffraction.

Large-angle scattering is due to the order in the loca­tion of atoms ins'ide the crystallites. Small-angle scatter­ing, on the other hand, is connected with the orderedarrangement of rather large formations that are calledpermolecular. It can even turn out that there is no orderat all inside these formations consisting of hundreds andeven thousands of atoms. But if such large systems form

3. Hard Electromagnetic Radiation 107

one-dimensional, two-dimensional or three-dimensionallattices, then small-angle X-ray scattering will tell thestory. To get a picture of what I mean, imagine a care­fully arranged structure made up of bags of potatoes.It is extremely interesting, and probably there is pro­found meaning in the fact that we encounter just such a"baggy" order in a very large number of biological sys­tems. For example, the long molecules that make upmuscle tissue are arranged very accurately, like a packageof round pencils. As small-angle X-ray scattering shows,we find just such extraordinarily high order in cell mem­branes, in such protein systems as viruses. and so on.

There is an interesting theorem in the theory of diffrac­tion. I will not try to prove it but I'm sure it will appearnatural to the reader. It can be rigorously demonstratedthat the type of diffraction pattern remains the sameif in the object that produces the diffraction we inter­change the apertures and opaque places. This theoremsometimes makes the research worker miserable. I t is whenX-ray scattering can be caused either by pores within thesubstance or by extraneous inclusions. The study of pores­their dimensions, shape, and quantity per unit volume­is of great practical interest. The colouring of syntheticfibres depends to a very large extent on these peculiaritiesin their structure. It is easy to see that an uneven distri­bution of pores will result in an uneven colouring anda final unpleasant fabric.

From what has been said it is now sufficiently obviousthat radiography of materials is not only a method ofstudying substances but also a method of technical con­trol in a great variety of industries.

1094. Generalizationsof Mechanics

Relativistic Mechanics

Newtonian mechanics that we discussed in the firstbook of this series is one of the greatest achievementsof the human genius. It made it possible to compute thepaths of planets, the trajectories of rockets and spacecraft,the behaviour of mechanisms. The development of phys­ics in the twentieth century demonstrated that the lawsof Newtonian mechanics have two limitations: they be­come unsatisfactory when dealing with the motion ofparticles of small mass and they fail us when we haveto do with the motion of bodies with velocities close tothe velocity of light. For small particles, Newtonianmechanics is replaced by the so-called wave mechanics;for fast bodies, it gives way to relativistic mechanics.

Classical mechanics is also made more complicatedwhen we encounter very great forces of gravitation. Theunimaginably strong fields of gravitation that govern thebehaviour of superdense stars force us beyond the limita­tions of such simple formulas of mechanics as were ex­plained in the first book. We will not dwell on suchchanges here but will take up the two highly importantgeneralizations just mentioned when dealing with themotions of microparticles and when studying velocitiescomparable to the velocity of light.

Let us begin with relativistic mechanics. The pathleading to this important division of physics least of allresembles a straight road. Not only is it tortuous but it

4. Generalizations of Mechanics

even passes through quite different countrie.s. The stayt~ngpoint here is the ether. Generally speakmg, phYSICIStSwere havina- it easy at the end of the 19th century. M~xPlanck's te'"'acher did not advise him to take up phYSICSbecause that science, he said, is practically complet~d.There were only two slight blemishes .on the ot~erwlse

perfect edifice of physics: drawbacks m accountmg forthe radiation of a black body (when that was cl~ared ';lp,physicists discovered the quantum). and the ~lstressmg

experiment of Michelson. This expenment, ~lllCh 'provedthat the velocity of light does not co~bme wlt.h thevelocity of the earth in its orbital mot~on and IS thesame in all directions, caused some anxIety about theproperties of the ether. .

Hardly anyone doubted the existence of some kmd ofdelicate matter whose oscillations constituted the electro­magnetic waves. After over a hund:ed years have passedit seems even surprising that despIte the gre~t numberof inconsistencies that the "ether" hypothesIs led to,the vast majority of scientists-some extremely talent­ed-went to great lengths to introduce no end of sup~le­mentary assumptions with th~ sole purpos.e .of salv.agmgthe concept of light as the motIOn of an mVlslble ultimatematerial sl1bstance. .

Some pictured the ether as a placid sea through whiChthe planets moved; others th.ought t.hat the ether coul.dbe entrained like air by movmg bodieS. As strange as Itmay now seem, nobody suggested the o~vious idea thatthe oscillations of electric and magnetIC vectors occurat a point, and for that reason cannot be accounted forin terms of mechanical displacements. Every effort wasmade to hold onto the old concepts, theories were setforth in which formally proper mathematical expressionswere derived (the infamous square root V1 - (V/C)2 includ­ed as well· here v was the velocity of a body and c thevelocity of light), but the equations were wrongly inter-

Albert Einstein (1879.1955) -the genius who created the theoryof relativity and revolutionized all physical thinking. In 1905,Eins~e~n published a treatise devoted to the special theory ofrelatIvIty. In 1907, he obtained a formula relating energy andthe ma~s ?f a body. In 1915, Einstein published his general theoryof relativIty. From this theory there followed new laws of gravita­tion and conclusions concerning the curvature of space.

Photons and Nuclei 110 4. Generalizations of Mechanics 111

preted. Particularly upsetting was the Michelson experi.ment, first carried out in 1881. Using an interferometer(described in Chapter 2), Michelson demonstrated thatthe velocities of light along and across the orbital motionof the earth are practically the same.

Even this fact (completely upsetting the theory con­cerning the existence of an ether) did not compel leadingphysicists to give up faith in a tenuous matter permeatingall bodies. Michelson's experiment forces us to give upany idea of an ethereal wind? Good and well, we willgive up the wind and the universe will be still morebeautiful with a motionless ether, and we will acceptthe Newtonian absolute space with respect to which allcelestial bodies move.

To account for the Michelson experiment, such outstand­ing physicists as Sir Joseph Larmor (1857-1942) andHendrik Antoon Lorentz (1853-1928) applied the hy­pothesis of the contraction of bodies in the direction oftheir motion. However, logical inconsistencies and theartificial nature of the explanations of many phenomenaconcerning electrodynamics continued to leave a senseof inadequacy.

The Gordian knot of contradictions was finally cut bythe greatest physicist of our century, Albert Einstein(1879-1955).

The starting point in Einstein's reasoning was theprinciple of relativity. There was hardly any doubt inanyone's mind, after Galileo, that in regard to mechani­cal motions, all inertial systems are equivalent (please

The theory of relativity does not exhaust Einstein's contributionto physics. From the research of Max Planck he drew the conclu­sion that there exists a light particle (called a photon) and demon­strated that from this standpoint it is possible to account for anumber of fundamental phenomena, including the photoeffect.

Photons and Nuclei 112

turn back to book one in this series and review the factsabout inertial systems). What appeared to be ratherstrange and esthetically unpleasant was that we haveequ.ivalence for mechanical motion but not for electromag­netic phenomena. Now suppose we assume the principleof relativity to hold true for all phenomena. But doesn'ttha~ lead to a cont~adiction? Imagine a source emittinga IIgh~ ra~. The lI~ht wave sets out into space and,observmg It, I permIt myself to disregard the fact thatthe source may be in motion or at rest. But if that is sothen the velocity of electromagnetic waves (299 792 km/s)~ust be the sa~e fro~ the viewpoint of observers livingm systems movmg wIth respect to one another with anarbitrarily high velocity v.

Suppose we have a train on a straight section movingat a con~tant speed.of ~. Parallel to the track is a highwayand on 1t ~nd movmg.m th.e same direction is a speeding~otor~y.clIst. A traffIc officer sees the culprit and withhI~ mInIature radar quickly finds that the motorcycle isdomg 85 ~m/hr. The locomotive engineer glances at themotorcyclIst and sees him catching up with the trainand then overtaking it. This observer has no difficulty~n ;neasuring the. spe~d of the motorcycle. It~s u =35 km/hr. It IS ObVIOUS that the speed of the trainIS 50 km/hr. The law of composition of velocities holdstrue:

u=v+ u'

T~is most obvious of all rules does not fit the caseof lIght rays. Photons move with the same velocity withrespect to two observers located in different inertial sys­tems.:rhe genius of Einstein lay in the fact that he rejectedt~IS. most obvious conclusion not only for light but,almmg to retain a unified approach to all physical phe­nomena (both electromagnetic and mechanical), was so bold

4. Generalizations of Mechanics 113

as to give up the law of composition of velocities for allbodies.

From this standpoint, there is no need to explainanything in Michelson's experiment. Since the velocityof light does not depend on the velocity of motion of thesource, it (the velocity) will be the same in all directions:both along the earth's orbit and across the orbital pathof the earth around the sun.

It follows immediately from the principles just for­mulated that the velocity of light is the maximum veloc­ity.* Indeed, if the velocity of light is never combinedwith the velocity of any source, that means it is impossi­ble to overtake light. In his reminiscences, Einstein recallsthat as early as 1896 he was considering the followingquestion: "If it were possible to set out after a lightwave with the velocity of light, then would we be dealingwith a time-independent wave field? That does indeedseem to be impossible."

To summarize, then: there is no particle that can movewith a velocity greater than the velocity of light. Let usgive some thought to this statement. It is so paradoxicalthat it needs repeating. If an electromagnetic wave takesoff from the earth or another planet in some direction,then the rate 'of propagation of that wave, when mea­sured by a terrestrial observer and by an observer flyingover the earth in a spacecraft moving with a fantastic

*Generally speaking, however, relativistic mechanics is not con­tradicted by the existence of particles moving with velocitiesarbitrarily greater than the velocity of light. Theoreticianshave even given them a name: they are called tachyons. However,if such particles existed, the velocity of light would still be~ limiting velocity for them-it would be the minimum veloc­ity not the maximum velocity. I personally think that thetheory of tachyons is nothing more than an elegant mathematicaltoy. If the world of tachyons did indeed exist, then events takingplace in the universe would, in principle, never be influencedby that world.

8-2542

Photons and Nuclei 114

velocity, is the same. This assertion holds true also forany particle moving with a velocity equal to the velocityof electromagnetic waves. Light is no exception in theEinstein theory.

Now what happens when the velocity of a moving bodyis less than the velocity of light? Obviously, in this casetoo the simple principle of the composition of velocitiesthat is so usual does not hold true. However, the devia­tion from the ordinary rule for composition of velocitiesbegins to be felt only when the velocity of the body isvery great.

Relativistic mechanics-that is the name given to themechanics of fast-moving bodies-leads to the followingrule for the composition of velocities:

v+ v'V= .1+~

c'

Make a rough estimate of what the values of v and v'must be for corrections to the simple rule for compositionof velocities to be needed.

How about spaceflight? Does the ordinary rule for thecomposition of velocities function when we deal withspeeds of tens of kilometres per second?

We know that a good way to gain speed is to launch asecond rocket system from a spaceship in flight. This couldbe one way of launching spacecraft to the outer reaches ofthe solar system. Denote by v the velocity of the space­ship relative to the earth, and by v' the velocity of thecraft launched from the spaceship relative to the space­ship. Suppose both velocities, v and v', are equalto 10 km/s. Now compute the exact velocity of the space­craft relative to the earth, using the exact formula ofthe composition of velocities. To the 1 in the denominatorwe have to add the fraction 102/(9 X101°)=10-9 • Thecorrection is practically negligible, which means the clas-

4. Generalizations of Mechanics 115

sicaI rule for the composition of velocities holds true.The question now is: When does relativistic mechanics

come into its own? We will have the answer soon, butmeanwhile let us continue the consequences that stemfrom the hypotheses just formulated. ?i.nce we are f?r~ed

to give up the principle of composlt~on of Vel?Cltle.s,we must be ready to introduce essential correctIOns 1Il

the other formulas of mechanics as well.As we have already mentioned, it was Michelson's

experiment that played a decisive role in the developmentof the theory of relativity. Michelson demonstrated con­clusively that the velocity of light along the earth'sorbit and across it is the same.

We will not consider the path of rays in the Michelsoninterferometer and will confine ourselves to simpler events.Somewhere on the earth a very simple laser experimentis set up on a tower at' an altitude l a~ove the. eart~ 'ssurface. The ultra fine laser beam goes 1Il the dnectlOnof the earth's radius, is reflected from a mirror placedon the earth's surface, is returned to where it startedfrom and is received by a photocell that engineers placein s~ch a position that we regard the light source andthe receiver of light to be in a single point. In Figure 4.1it is labelled S. Using an ultrarefined stopwatch, wecan fix two instants: the first :when the light set out andthe second when it reached the photocell.

Two observers follow this event. The first one standsnext to the experimental setup, the other one is perchedon a distant star. Both observers measure the time inter­val -r between the two events: the emission of light andits return to point S. The first draws the path ?f the rayand it turns out to be simplicity itself. He belIeves thatthe paths of the ray there and back coincide absolutely.His reasoning is corroborated by the equation -r=2l/c.

The stellar observer follows the flash as the light startsout on its journey and records its arrival at the photocell.8*

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Figure 4.1

116 4. Generalizations of Mechanics 117

The time interval that he measures is equal to 'to To besure everything is properly done, he too draws the pathof the ray. For him, however, the positions of point S atthe time the stopwatch was switched on and the instanthe recorded the reaction of the photocell do not coincide.The path of the light ray in his case is different. Thestellar observer knows the velocity of the earth relativeto himself and so what he gets is an equilateral triangle,the base of which is equal to v't and the altitude to Z.Using the Pythagorean theorem, the stellar observer findsthat the path traversed by the light ray is equal to2 VZ2 + (V't/2)2. This path is equal to C't, because the ve­locity of light is the same for all observers. If that is so,the time interval between the two instants is equal to

2l't=----===

c 1/ 1 __1>_'r c'

That is a surprising result I From the standpoint ofthe terrestrial observer, the same time interval betweenthose very same events is equal to 2l!c.

Appealing to logic, we can draw an unavoidable con­clusion: the time reckoned by an observer at rest differsfrom the time reckoned by an observer in motion.

The time of the stationary observer is called the prop­er time and is denoted by 'to. We find that the time ofan observer moving with velocity v is connected withthe proper time by the expression

to v't = 11 ' where 13 = -

1 -132 C

What this means is that a moving clock goes more slowlythan a stationary clock. If we accept the basic postulatesof the theory, we cannot escape such a conclusion. Andthis leads to the strange, at first glance, consequencethat the idea of simultaneity has to be given up.

m = -.,-,=m=o=-V1- /32

I t is quite natural that the mass increases with anincrease in velocity. Indeed, if velocity has a limit, thenas a particle approaches that limit, it becomes progres­sively more difficult to accelerate the moving particle.That is precisely what is meant when we say that themass of the particle increases.

For a long time, no one encountered such high veloci­ties that would force one to take into account the differ­ence between the square root and unity in the formulasfor distance and time. It was only just recently thatthe truth of the formula for time was demonstrated.

Now about the dependence of the mass on the velocity,it was revealed in the case of a stream of electrons evenbefore Einstein's article appeared. The formula for themass is an engineering formula in the full sense of theword. As we shall see in the next section, without itone simply could not even design a modern particleaccelerator. In these very expensive machines, nuclearparticles are accelerated to such an extent that the squareroot gets much closer to zero than to unity.

The formula illustrating the dependence of mass onvelocity was first proposed before Einstein. But beforethe advent of relativistic mechanics its interpretationwas not correct.

However, the famous expression E=mc2 , which connectsmass and energy, was derived by Einstein. That formulaand also the dependence of 1, 't, and m on the velocityfollow rigorously from the postulates of the theory.

Multiply the mass by the square of the velocity oflight. For a moving body it is mc2 , for the same bodyat rest it is moc2 • Let us form the difference of these twoexpressions:

mc2 - m c2 = m c2 ( 1 - 1)o 0 \Jlt~IF

Photons and Nuclei 118

But won't it turn out, then, that from the viewpointof one observer, Jim fired a gun and John was killed bythe bullet while, from another point of view, John waskilled first and Jim fired later? The reader can restassured that relativistic mechanics does not allow forany such nonsense. The principle of causality will neverbe upset. And the proof can even be given popularly,but, unfortunately, that is beyond the scope of this book.

However, a few words are in order about the so-calledtwin paradox which even now is brought forth as proofof the inconsistency of the theory. John and Pete aretwins. Pete takes leave of John and sets out on a spacevoyage with a velocity close to the velocity of light,and after a certain time returns to earth. Pete's clockhas been going slower and so he returns to earth stillfresh and young, whereas his brother John has grownold and feeble.

However-fortunately or unfortunately, depending onwho likes what-no such meeting can be organized, thatis, if we keep to the conditions under which the formulasunder discussion hold true. The point is that Pete wouldhave to reverse his velocity, and so the conclusions re­ferring to inertial systems have nothing to do with thiscase.

The relativity of time is not the only consequence ofthe new theory. Just as the proper clock of the observergoes faster than any other clock, so also the length ofa rod, 10 , held in one's hand is the maximum length.From the point of view of any observer moving withvelocity v along the rod, that same length is equal to

lo~/1- ~2.

The same root also appears in the expression for themass. The mass mo of the body that the observer holdsin his hands is termed the rest mass. It. i~ minimum.For a moving observer,

4. Generalizations of Mechanics 119

Photons and Nuclei 120

Taking advantage of the approximate equation, the truthof which you can verify with ease, we have

1 -1 + ~A2V1- /32 - 2 ....

The difference that we are calculating is of the formm v2

mc2 - moc2 = --t-As you can see, it is equal to the kinetic energy of the

body.Thinking about this equation, Einstein came to the

following fundamental conclusion. The energy of a mov­ing body may be expressed as

E=mc2

This energy is made up of the rest energy moc2 and theenergy of motion. Without having any information aboutthe structure of a body and not knowing the nature of theinteractions of its component particles, we can state thatits internal energy is equal to

U=moc2

The internal energy of a body of mass 1 kilogram isequal to 1017 joules, which is just the amount of heat thatwould be released if we burned three million tons ofcoal. As we will soon learn, physicists have been ableto release a small portion of this energy by smashingheavy atomic nuclei or forcing light nuclei to coalesce.

It must be stressed that the Einstein equation, E=mc2 ,

has to do not only with the energy inside nuclei. Theequation is universal. But this is just like the spacemen'sclocks. For the most part, the relationship between energyand mass cannot be verified. Indeed, take a ton of molyb­denum and heat it to 1000 degrees. The mass will increaseby 3 millionths of a gram. Only the fantastic magnitudes

4. Generalizations of Mechanics 121

of intranuclear forces enable us to grasp the correctnessof the equation E=mc2 •

This is a good time to warn the reader about sloppystatements of this remarkable equation. Some say thatmass turns into energy; or, still worse, that matter isconverted into energy. Actually, the formula E=mc2

states that whatever mutual transformations of differenttypes of matter occur, the change in energy that takesplace in the system corresponds to an equivalent changein mass. Energy and mass are two uniquely related charac­teristics of matter.

Particles with VelocitiesClose to the Velocity of Light

The desire to reach down to the elementary buildingblocks of matter that make up the world is as old as theworld. But for many long centuries this subject was atthe mercy of the scholastic reasoning of wisemen. As soonas the opportunity appeared of actually breaking up mol­ecules, atoms, and atomic nuclei, physicists took up thechallenge with inspiration and persistence. The work goeson today and, to be quite frank, there appears to be noend in sight.

Obviously, to find out what the world is made of, onehas to break up the component particles. That requires"projectiles", and the more energy they have the greaterthe hope of penetrating that secret of nature.

The history of the generation of fast particles beganin 1932 when Rutherford's coworkers constructed a de­vice to produce protons, which were accelerated to ener­gies up to 500 kiloelectron volts. Then came cyclotrons~apable of racing protons to energies that were measuredIII megaelectron volts (mega means a million). Next camethe synchrotron that accelerated protons to a thousandmillion electron volts, and the era of giga volts was here

Photons and Nuclei 122

(giga means a thousand million). Today, machines arebeing designed that plan to reach energies of millions ofmillions of electron volts. At a conference of physiciststhat took place in 1975 at Serpukhov (USSR), which hasone of the biggest machines of that type, construction ofa circular machine 16 kilometres in diameter was suggest­ed.

The reader of course is already asking the obviousquestions of how these machines work, why they are cir­cular, and what the idea is all about anyway.

Actually, any vacuum device with a high voltage ap­plied to the terminals can act as an accelerator of particles.The kinetic energy of particle accelerated to high velocityis equal to

mv2

-2-= eU

(we have been making so much use of this formula thatthe reader has most likely got it firmly in mind by thistime, which is all to the better). Even X-ray and televi­sion tubes can be regarded as accelerators.

But that principle of acceleration does not yield highvelocities. The term "accelerator" is applied to machinescapable of accelerating particles to near-light velocities.To attain that end, the particle has to be accelerated insuccession through a number of fields. It is easy to seethat a linear accelerator is not very convenient becausea good many centimetres of path length is needed forsome miserable tens of thousands of electron volts. Inthat way, reaching 10000 million electron volts wouldrequire a path length of ten or so kilometres.

No, such a brute force approach to the problem is notsuitable. In 1934 the American physicist Ernest OrlandoLawrence (1901-1958) laid the foundation for the construc­tion of modern circular accelerators, which he christenedcyclotrons. A single machine combines the acceleration

4. Generalizations of Mechanics 123

of a particle by an electric field and its return many timesto the original interval (gap) with the aid of a magneticfield.

The Lawrence accelerator looked like a tin can cut intwo along a diameter. A rapidly alternating voltage isapplied to the two halves, and the charged particles are ac­celerated during the time intervals in their traverse of thegaps between the halves. Inside the "tin can" we makethe particles move in circular orbits by imposing a mag­netic field whose lines of force are perpendicular to thebottom.- In that case, as we know, the charged particledescribes a circle of radius

R=~eH

The period of revolution is equal to

T = 27tmeH

For the electric field at the gaps of the machine to beable to pick up the particles, the variable voltage mustbe timed so that it changes sign at the very i~stant theparticle arrives at the gap between the halve~.

The charges are generated in the centre of the machine(for instance, ionization of hydrogen generates protons).The first circle is of small radius. But each successivecircle increases the radius since, by the formula we justgave, it is proportional to the velocity of the particle.

At first glance it might appear that by increasing thedimensions of the cyclotron, and with it the radius of thecircular trajectory, we could impart any desirable energyto the particle. And when the desired energy is attained,all we have to do is direct the beam out of the circle bymeans of a deflecting plate. This would be an ideal setupif it weren't for the dependence of mass on velocity.Einstein's equation for mass that did not seem to have

4. Generalizations of Mechanics 125

reader that that is precisely the situation (specify therate of increase in the strength of the magnetic field,compute the particle flight path, and then draw the curve,and you will see for yourself how the principle of phasestability functions). Or you can take my word for it thatin this way we can, in principle, increase the velocityof particles without bound. We will have to use the pulsemethod of acceleration however. The machine operateswhen the field is increasing and is idle when in reverse.But we will not dwell on this method here. It too is outof date. If we kept to that principle, modern acceleratorswould have steel magnets weighing millions of tons.

Modern circular accelerators, called proton synchrotron,accomplish particle acceleration in a single orbit, andso the whole central portion of the magnet is cut out,as it were. These machines also operate on the pulse meth­od. Here, both the intensity of the magnetic field andthe period of the electric field vary in synchronism.Suitable particles build up speed as they move in a strict­ly circular orbit. The less suitable particles will oscillateabout the good orbit but will also acquire a certain veloc­ity.

In principle, fantastic accelerations can be achieved.For instance, proton velocities just short of the velocitylight have been attained.

Now comes the question of why such machines areneeded. Particle accelerators are constructed to learn moreabout the physics of elementary particles. The higherthe energy of the charged particles used as projectilesto bombard targets the greater the chances are of dis­covering the laws of the mutual transformation of suchelementary particles.

Roughly speaking, the world is made up of only threeparticles: electrons, protons, and neutrons. So far, theelectron has no justification for its status as a componentparticle. As for protons and neutrons, it turns out that

124

the basis for designing

Photons and Nuclei

any practical value now becomescircular accelerators.

Since. the m~ss of a particle increases with velocity,the orbItal perIod does not remain constant, it increases.The particle begins to come late and arrives at the accel­erating gap not when the voltage phase changes 180 de­g;ees but later. As the velocity increases, there comes atIme when the electric field not only ceases to pick upthe particles but even begins to slow them down.

A cyclotron permits speeding protons to about 20 mega­el~ctron volts. Actually that is not very much. As I havesaId before, physicists are asking for more powerful ma­chines. Obviously new approaches to the problem areneeded if high energies are to be attained.

And the for~ o~ the equation for the orbital periodsuggests the dIrectIOn to take. The mass increases withincr~asing.velocity. Then what we need to do is increaseth~ IntenSIty of the magnetic field apace in order to main­t~In the period. ~rue, this is a simple solution only at~rst glance. Don t forget that the radius of the orbitIncreases with each circuit of the particle. So it is requiredthat the synchronous increase in mass and magnetic fieldhold go?d for a particle making successive circuits ofprogressn:ely i?creasing radii. If we can disentangle thisInterrelatIOnshIp of quantities, it will be clear that, firstof all, there are "suitable" particles for which the condi­tion will be fulfilled for a certain specified rate of increasein intensity ~f the magnetic field. What is most impor­tant, the~e WIll occur a peculiar kind of phase stability~automatlC phase stabilization). A particle whose energyIS greater than the energy needed for its orbital radiuswill slow down due to the excessive increase in mass'a?d, contrariwise, a lack of energy will lead to accelera~tiOn.

Very simple calculations using the formulas for theradius and orbital period of a particle will convince the

Photons and Nuclei 126

they can be broken up into pieces. New particles arise indiverse collisions that occur among the fragments. Todaythere are something like 250 such particles, and the great~ity is ~hat their number is constantly increasing (with thelllcreaslllg power of accelerators). Specialists in the fieldof elementary-particle physics hope to develop a table ofelementary particles like the Mendeleev table of elementsand then reduce them to a small number of what might becalled "protoparticles". After all, haven't the hundredand so chemical elements and several hundred isotopesbeen reduced to combinations of electrons, protons, andneutrons?

The read.er then has every reason to'ask: What meaningdo we attrIbute to the phrase that the world is made upof three particles? The point is this. Only the protonand the electron are absolutely stable particles. The neu­tron is not quite stable, if the word "stable" is taken in its~veryday usage .. B~t its lifetime in the world of particlesIS tremendous: It IS equal to approximately 103 seconds.As for the multitude of other elementary particles thatare such a worry to theoreticians, their lifetimes comeout to less than a millionth of a second. This is of coursea very big difference.

Nevertheless we would like to bring these short-livedfragments of matter into some order. Many types of sys­tems have been proposed for the elementary particles.But as soon as a more powerful accelerator comes intoaction and fresh phenomena are observed, the oldschemes have to be discarded.

At the time of writing, I see that the specialists areoptimistic. The entire system of elementary particles isbeing reduced to certain "protoparticles" called quarks.The only trouble is that quarks, unlike electrons andprotons, have never been observed and, what is more,probably (in principle) never will be. In order to setup a "Mendeleev" table of elementary particles, the quark

4. Generalizations of Mechanics 127

must be endowed with an electric charge either one thirdor two thirds of the electron charge,and it must havetwo additional parameters that are quite unimaginable.These parameters are called strangeness and charm*.

I will not go into the problems that plague the ele­mentary particles, and not because it is difficult to givea popular version of the existing schemes but simplybecause it is too early yet to be sure of their charm, soto say. It might very well be that entirely new ideas arein the making pertaining to elementary particles andquite new principles of approach to these minute portions(measuring in centimetres one over unity followed byfifteen zeros) of the universe.

Wave Mechanics

In 1923, in a scientific paper of exceptional boldnessand with a streak of genius the French physicist LouisVictor de Broglie (1892- ) wrote: "In optics, over the cen­turies, the corpuscular approach to phenomena has beenfar too neglected in comparison with the wave approach;might it not be that in the theory of microparticles thereverse mistake has been made?" De Broglie pointed outthe path to follow in order to relate particles with waveconceptions.

His work was continued and expanded by the marvelousAustrian physicist Erwin Schrodinger (1887-1961). Some­what later in 1926-1927, it became clear that thewave mechanics and quantum mechanics are actuallyequivalent terms. This new mechanics is one of the mostimportant branches of physics that teaches us how toregard the behaviour of microparticles when neither their

*Since the manuscript of this book was handed in, new eventshave taken place in the world of elementary particles, a newparameter colour has appeared.

128

aspect suffice to inter-

Photons and Nuclei

corpuscular aspect nor their wavepret events.

We warned the reader not to take too literally theexpression "electromagnetic wave". Radio emission, light,and X rays can all be regarded in two aspects: wave andparticle (corpuscular). The very same holds true forfluxes of particles. Although particle fluxes have clear-cutpeculiarities that distinguish them from electromagneticradiation (the main being that electrons, nuclei, neutrons,and ions can have a whole range of velocities, whilephotons can only move at a velocity of 300 000 km/s),this kind of matter also exhibits wave properties in cer­tain situations and corpuscular (particle) properties inothers.

What is the wavelength that must be attributed to amoving particle? By means of the following (somewhatsimplified) reasoning, de Broglie demonstrat6d (rather,conjectured) what wavelengths are associated with whatparticle fluxes.

Let us examine the basic relations that connect theparticle aspect of electromagnetic radiation with the waveaspect. A portion of energy of electromagnetic radiationcarried by a photon is given by the formula E=hv. Theenergy of a photon, like that of any other portion ofmatter, obeys Einstein's equation. Thus, the photon ener­gy can also be expressed by the equation E=mc2

• Fromthis it follows that the mass of the photon is m=hv/c2

•

Multiplying together the mass and the velocity, we obtainthe value for the momentum of the photon as*

hv hp=-=-

c t..

*The mass of the photon is the mass of a moving particle; nowthe rest mass of the photon is practically zero; experimentersclaim it is less than 0.6 X 10-20 megaelectron volt. Also notethat the relation for the momentum of the photon may be veri­fied directly by measuring the pressure of light.

4. Generalizations of Mechanics 129

But we want to know the wavelength of a particlewhose rest mass is different from zero. What can thatbe? Suppose the foregoing reasoning holds; assume thatthe relationship between the momentum and the wave­length is universal! It then remains to rewrite this ex­pression in the form

A. = _h_mv

This is the famous de Broglie equation (or relation).It shows that the wave aspect of a flux of particles mustbe revealed with particular clarity when the mass andthe velocity of the particle are small. This is confirmedhy experiment because it is easy to observe the diffractionof particles in the case of electrons and slow neutrons.

Verification of the truth of the foregoing, which inci­dentally in its day was regarded as a play on concepts,is quite straightforward. From the same substance, takean X-ray photograph (roentgenogram), an electron diffrac­tion pattern, and a neutron diffraction pattern. If theparticle velocities are matched so that the wavelengthsin all cases are the same, then we should obtain identical(with regard to the radii of the rings) Debye crystallo­grams. And that is precisely what happens.

In 1927, an accident brought about the first verificationof the de Broglie relation. Two American physicists,Clinton Joseph Davisson (1881-1958) and Lester HalbertGermer (1896- ), were conducting experiments in the scat­tering of electrons on the surface of metals and, whenworking with their im;trument, they happened to heatup the object. The object was a polycrystal; after beingheated it was recrystallized, and so now the rays werescattered by a monocrystal. The pattern obtained was somuch like appropriate roentgenograms that there was nodoubt that electrons are capable of diffracting just likeX rays.9-2542

Photons and Nuclei 130

Rather soon, observations of electron diffraction turnedinto a method of investigating the structure of sub­stances, and in many instances it was more suitable thanX-ray.diffraction analysis. The main application of elec­tron d~ffraction analysis is the stud v of the structure ofthin films. The principles don't d{ffer at all from theprinciples we discussed in Chapter 3. The difference liesin the fac~ that electron rays are scattered by electronsand nucleI, whereas X rays are scattered by electronsalone.. Since the wavelength of a particle is inversely propor­

tIonal to the mass, it is clear that the diffraction ofmolecules is observed with difficulty. At any rate, ith~s not. been achieved yet. It is possible to observe thedIffractIOn of protons, but it is of no interest: protons donot suffice for studying volume structure due to thl' lowpenetrating power, and for surface studies it is betterto use the diffraction of electrons because it yields farmore information about the structure.

With neutrons it's different. Research into the diffrac­tion of these particles has become a point of interest tohundreds of scientists. It is caJJed neutron diffractionanalysis.

It is technically much more difficult to obtain a neu­tron diffra~t~on pattern than a roentgenogram. First ofall, a suffiCIently strong flux of neutrons of suitablewavelength (wavelength is regulated by neutron velocity)can only be generated by taking these particles out ofa nuclear reactor through a special channel. The seconddifficulty. is that neutrons are not easily scattered; theyc~n readIly pass through a substance without collidingw~th the atomic nuclei. That makes it necessary to workwIt.h large crystals of the order of a centimetre in size.I ~ I~ not e.asy to obtain such crystals. Finally, the thirddIffIc~lty IS that neutrons do not leave traces in a photo­graphIC plate and reveal themselves in ionization cham-

4. Generalizations of Mechanics 13'1

bel'S only indirectly. How neutrons are counted will bediscussed later on.

Then why are scientists interested in neutron diffractionanalysis? The point is that neutrons, unlike X rays, arenot scattered by electrons but are deflected from theirpaths in encounters with atomic nuclei. There are manysubstances whose atoms differ only slightly as to numberof electrons but differ radically as to properties of thenuclei. In such cases, X rays cannot distinguish theatoms but neutron diffraction analysis is highly success­ful. But perhaps the most important of all is that neu­trons are strongly scattered by the nuclei of hydrogenatoms, whereas X rays can establish the positions ofhydrogen atoms only with difficulty: the point is thehydrogen atom has only one electron. Now it is veryimportant to know the position of this atom. In numerousorganic and biological systems the hydrogen atom bindstogether the parts of a single molecule or adjacent mole­cules. This particular bond is called the hydrogen bond.Again, neutron diffraction analysis reigns supreme indistinguishing atomic nuclei having distinct magneticproperties. All these factors are sufficient to make neu­tron diffraction analysis an important method of studyingthe structure of substances.

The Heisenberg Uncertainty Principle

For a long time, many physicists could not reconcilethemselves to the fact that light and particles simultane­ously possess wave and corpusc?lar propert~es. This dual­ism seemed to contain somethlllg contradICtory to thetheory of knowledge. In particular, these scientists wereopposed to the Heisenberg p~i~ciple.. .

This most important propOSItIOn of P~YSIC~ .of the mI­croworld establishes the range of applIcabIlIty of thecorpuscular aspect of any phenomena connected with the

Photons and Nuclei 132

motion of particles of a substance. The Heisenberg uncer­tainty principle can be written as follows:

Ax Av>..!!-m

Here, Ax and Av are the "smeared nature" of our knowl­edge of the coordinate and velocity of motion (in thedirection of the same coordinate axis) of a blob ofmatter that we are considering in the corpuscular aspect.Briefly, Ax and Av state the indeterminateness in ourknowledge of the coordinate and the velocity of the par­ticle.

It must be stressed at this point that we are not speak­ing of the technical difficulties of measuring. The fore­going relationship relates uncertainties that cannot beeliminated even in the most ideal experimental setup.Today, the various schemes proposed for an absolutelyexact measurement of the trajectory and velocity of mo­tion of particles are only of historical interest. A carefulexamination of them will always turn up a fundamentaldefect in the proposed scheme.

I will attempt, in a few words, to explain why experi­ment cannot yield greater accuracy than the Heisenbergprinciple allows for. Suppose we are discussing a definiteposition of a particle in space. To learn where it is locat­ed, the particle must be illuminated. As has already beenmentioned, the possibility of distinguishing details isdetermined by the wavelength of the radiation used.The shorter the wavelength the better. But as we dimin­ish the wavelength, so we increase the frequency of thelight, and, hence, also the energy of the photon. Theimpact experienced by the particle under study will makeit impossible for us to judge the velocity it had at thetime of its encounter with the photon.

Or take this classical example. We place a narrow slitin the path of an electron. The electron passes through

4. Generalizations of Mechanics 133

the slit and falls on a screen. A flash is seen. Thus, towithin the accuracy of the width of the slit, we haveestablished the position of the electron at the time itwas passing through the aperture. Let us improve theprecision. For this purpose we reduce the size of the slit.But then the wave properties of the electron will beginto take over (see page 57). The electron can deviate fartherand farther away from the straight path, and that meansthat we will progressively lose more information aboutthe component of its velocity in the direction of the planein which the slit is made. Dozens of such cases canbe thought up and we Can consider them quantitatively(which is exactly what was done in the 1930s), and everytime we arrive at the above formula.

Let us now discuss the estimates of Ax and Av thatcan be made with respect to particles of different massby using the Heisenberg inequality.

Suppose we are discussing an electron residing in anatom. Can we devise an experiment capable of establish­ing the site occupied by the electron at a given instant?Since the dimensions of an atom are of the order of 10-8 cm,we would like to obtain an accuracy of, say, 10-9 cm.In principle (true, only in principle) such an experimentis possible. But then let us estimate (using the inequality)the loss of information concerning the electron. For anelectron, him is roughly equal to 7 cm2 /s; for it, theHeisenberg uncertainty principle is written thus: Ax Av>>7. And so Av exceeds 7 X 109 cmls, which is absolutelymeaningless; in other words, we can't say anything aboutthe velocity of the electron.

Well, and suppose we try to make a more exact estimateof the velocity of the atomic electron. For this purposewe can even think up an experiment that can actuallybe performed. But then we will have lost all knowledgeabout the site where the electron is located.

The inequality applied to an atomic electron shows

Photons and Nuclei 134

that the corpuscular aspect does not operate in this case.The concept of the trajectory of an electron becomesmeaningless and nothing at all can be said about thepaths of transition of an electron from one energy levelto another.

The situation changes when we seek the motion of anelectron in ionization chambers. The track left by anelectron is visible. So there is a trajectory after all. Butthen how is this linked up with the foregoing calculations?The answer is: there is no connection. One merely beginsto reason anew. The track has a thickness of about 10-2

centimetre. This means the uncertainty with regard tovelocity even for a slow electron going through a chamberat about 1 kilometre per second is quite acceptable. It isequal to 7 metres per second.

These numerical examples show us that the corpuscularaspect begins to vanish as soon as we attempt to lookcloser and inspect the portion of matter in more detail.

We can very often speak of protons and neutrons asparticles, but if one is interested in their behaviour insidethe atomic nucleus, which is about 10-13 centimetre insize, then the corpuscular aspect is not revealed.

And it is not hard to image that we can easily visualizethe behaviour of a large molecule with molecular mass ofthe order of a million as if it were a pea. Such ·a moleculebehaves like a good and honest particle. It is even possibleto trace the trajectory of its thermal random motion.

The time has long since past when wave-particle dual­ism was regarded as something strange that requires aprofound interpretation. Such outstanding scientists asEinstein and Bohr argued vociferously about how tointerpret the "strange" behaviour of electrons and otherparticles. Today, the overwhelming majority of naturalscientists see nothing particular in utilizing the two as­pects in their descriptions of phenomena in which elec­trons, nuclei or photons participate.

4. Generalizations of Mechanics 135

Some ten years ago, a group of experts studying thework of scientists sent a questionnaire to a large.group ofphysicists (about ten thousand). One of the questIOns ,:as:Is the problem of the two aspects of matter of topIcalinterest and has it been completel~ explore~? Only.twentyreplied that they presume the H81senb.erg mequaht~. andits associated problems do not constitute the ultImate

truth. . II' .The difficulty in coming to terms WIt 1 t liS Important

law of nature was probably due to .a logical er~or under­lying a protest that might be stated 1Il the followmg words:"I cannot agree that the behaviour of particles of matteris unpredictable." The fallacy of tl~is sentence l.ies inthe fact that this parcel of matter IS spoken of m theordinary everyday sense of the word. Actually, the parcelof matter (we are speaking of light,. micro~a:es, el~ctronsor nuclei) is in no way like a gram. It IS Impo.ssible tovisualize a particle of matter; surely everyone wIll argee.Suffice it to say that to the electron or proton we cannotattribute colour, hardness, temperatur.e ... : All the~eproperties have to do with macroscopIC bodIes. Now ~fit is not possible to visualize a parcel of matt.er of t~ISkind, all the more is it impossible to conceIve of. ItSmotion. The motion of such a parcel of matter com bmestwo aspects the wave aspect and the corpuscular aspect.Therefore, ~nly one of these aspects is unpredictab~e.

Quantum mechanics (and, to repeat, wave mechalllcsis a synonym) gives us a set of rigorous rules by meansof which we can predict the behaviour of parcels of matter.A description of particles by the methods of quantummechanics gives an exhaustive representation of the. reg­ularities of the microworld. With its aid we unerrmglypredict events and compel them to serve us practically.

This of course does not mean that new and more generallaws of nature will not be discovered at some future timeand that quantum mechanics may then become a partic-

•

Photons and Nuclei 136

ular instance of such generalities, just as Newtonianmechanics became a particular case of more general me­chanics. Such general laws must be suitable for describingthe b~~aviour of particles of small mass moving at highvelocItIes. We are eagerly awaiting (and have been forquite some time) the creation of a theory that wouldunite all the "mechanics" into a single whole. There isalready a name for this as yet "uncreated" theory; it iscalled relati vistic quantum mechanics.

Is it not remarkable that the avalanche of discoveriesmade in the first quarter of the twentieth century hassuddenly come to a halt? The reader may find it strange.But the fact remains. Despite the fantastic progress ofapplied sciences and despite the fact that the next twoquarter-centuries have passed in the form of a scientificand technological revolution-despite all this, no newlaws of nature have been opened up after the discoveryof quantum mechanics. We will simply have to wait.

5. The Structureof Atomic Nuclei

Isotopes

In the third book of this series we told the story ofhow a flux of particles with different charge-to-massratios can be separated by means of electric and magneticfields. And if the charges are the same, then separationof particles is possible as to mass. This is done by a devicecalled a mass spectrograph, which is widely used forchemical analysis.

A diagram of this instrument is shown in Figure 5.1.The underlying idea is as follows. Particles with differentvelocities enter the electric field of a capacitor. Imaginea group of particles with the same elm ratio. A fluxof these particles enters an electric field and is separatedinto faster particles that exhibit a smaller deviation inthe electric field and slower particles with a greaterdeviation. In fan-like fashion, these particles then entera magnetic field perpendicular to the drawing. It is con­nected so as to deflect the particles in the opposite di­rection. Here too, the faster particles are deflected lessthan the slower particles. From this it follows that atsome point outside the field, the imagined flux of identicalparticles will again collect in a single point, that is tosay, they will come to a foc1ls.

Particles with a different elm value will also collectin a point, but that will be a different point. Calculationsshow that the foci for all elm values will lie very closeto a certain straight line. Now if a photographic plate

is positioned along this straight line, particles of eachkind will reveal themselves by a separate line.

Isotopes were discovered with the help of a mass spec­trograph. The honour for the discovery of isotopes goesto Sir Joseph John Thomson. In 1913, while studyingthe deflection of a beam of neon ions in an electric fieldand a magnetic field, he noticed that the beam was sepa­rated into two parts. The atomic weight of neon wasknown with sufficient accuracy: 20.200. It was foundthat in reality there are three kinds of neon atoms, withatomic weights 20, 21, and 22. Sorry, I'm rather used tothe old terms; these numbers are now called relativeatomic masses.

Since the chemical properties of neon do not dependon their masses, physicists were soon confident that thedifferences are connected only with the nucleus. Thecharge of the nucleus and the number of electrons arethe same and so different kinds of atoms of neon shouldoccupy the same place in Mendeleev's periodic table ofelements. Whence the name, isotopes, or those that occu-py the same place. .

In the 1920s, the mass spectrograph took on Its present-

1395. The Strudure of Atomic Nuclei

day features and a study began of the isotopic compo­sition of all elements. All elements, without exception,constitute a mixture of isotopes. There are some, likehydrogen and oxygen, that consist mainly of one isotope(hydrogen of mass 1-99.986%, oxygen of mass16-99.76%). There are others that contain different iso­topes in about equal quantities. Say, chlorine (75 % isan isotope of mass 35 and 25 % is an is?tope of mass 37).There are still other elements that conS'lst of a large num­ber of isotopes. The examples we have given are thoseof stable isotopes. Radioactive isotopes of an element(these are not stable and decay) will be discussed lateron. .

The mass spectrograph was soon refined t? the pomtwhere it was established that the masses of Isotopes areexpressed by whole numbers only up t? the. se~ond ~ofourth decimal place. The reasons for thIs devIatiOn willbe taken up later on.

The mass of an atom rounded off to a whole numberis called the mass number.

Since the nuclear mass does not affect the chemicalbehaviour of an element, there are obviously many ~h:m­ical compounds that differ in their isotopic composl.tiOn.For example, there are two kinds of water, or?marywater and heavy water. In ordinary water, we find anisotope of hydrogen with mass number 1, in heavy water(called deuterium), the isotope of hydrogen has massnumber 2. In the natural state, we find three isotopes ofoxygen (mass numbers 16, 17, and 18), w~ich.meansthat water is a mixture of molecules of SIX differentkinds. If the molecules of a substance consist of a largenumber of different atoms, then the number of isotopicvarieties may run into the tens and hundreds:

SeparatinO' isotopes is an important branch of mdustry.And it is pa~ticularly important i~ a number of processesinvolved in the generation of atomIC energy. Heavy water

138Photons and Nuclei

Figure 5.1

LASER BEAM

Figure 5.2

Hi

COLLECTOR

00\1EXCITED ATOM

ffi t Iy:::::::::::/ .---';;"~ ~(o 0

0

.o.c;MJ.,.~O.o.tP~iii'e04'ci~ .-:-;;'~ • ~og.O o.ATOMIC BEAM .". """[it\

FURNACE NONHOMOGENEOUS ELECTRICOR MAGNETIC FIELD

5. The structure of Atomic Nuclei

by means of a laser. A gas made up of atoms or moleculesemerge from an opening in a furnace. The laser beamexcites the atoms of one isotopic variety. As a rule, theexcited atoms possess an electric or magnetic moment,and so a nonhomogeneous magnetic or electric field willdeflect them to one side (upper diagram).

A second method is used if the excited atoms are rap­idly de-excited. In this case, the same atom passesthrough the space covered by a laser beam and is exciteda second time, thus several times experiencing inelasticcollisions with photons. Each absorption of a photonimparts to the atom a momentum that is in the directionof the action of the laser beam. Atoms that can be excited

Photons and Nuclei 140

has to be separated from ordinary (light) water, differenttypes of atoms of the nuclear fuel (uranium and thorium)have to be separated as well. And there are many manymore such industrial problems that confront the physicist.

The complexity of the problem is that the atoms exhib­it extremely .small .differen~es in their electronic makeupand, hence, m their chemICal properties. In the case oflight atoms, the separation procedure is carried out withextreme difficulty in the form of a multistage chemicalextr~ction pro~ess. In the case of heavy atoms, the onlypossIble way IS to apply physical methods that makeuse of slight differences in the masses of atomic nuclei.

. The. most widely employed method today is the gaseousdIffusIOn method. Molecules containing isotopes of differ­ent mass differ slightly as to their rates of passagethrough a porous barrier. Light-weight molecules getthrough faster than the heavier varieties.

Of course, the separation procedure could be based ont~e principle of the mass spectrograph that we have justdIsc~ssed. But both methods are time- and money-con­summg.On~y just two or three years ago it was demonstrated

that Isotope separation can be handled by a new lasermethod. The most important advantage here is that thelaser generates a ray of extremely high monochromicity.Now .the difference in. distances between energy levelsoccupIed by electrons m two isotopic varieties of thesame element is naturally very slight. This is due tothe mass. differe~ce of the nuclei since the charges onthe nuclei of two Isotopes are the same. And it is preciselythe charges th~t determine, on the whole, the locationof the electro~llc levels. Now a laser beam is so strictly~onoclll'?matlC that it is capable of exciting only onekmd of Isotope while leaving atoms of the other varietyunexcited.

Figure 5.2 depicts two processes for separating isotopes

Photons and Nuclei 142

are simply pushed upwards whereas atoms of the varietythat does not absorb photons are not deflected in theirmotion.

The first successful experiment of this kind was carriedout with a beam of barium atoms irradiated with a laserbeam of wavelength 0.55535 micrometre. Absorption ofa single photon shifted the atom through 0.8 centimetreper second in the case of a longitudinal velocity of50000 cm/s.

Radioactivity

In ~he third book of this series a short descriptionwas gIven of how Rutherford established that the atomconsists of a minute nucleus and of electrons movingaround the nucleus. Now comes one of the most importantchapters in physics. It describes the structure of theatomic nucleus, which is made up of protons and neutrons.Strange as it may seem, the history of this discoverybegan fifteen years before Rutherford demonstrated hisnuclea~ model of the atom in experim!'lnts devoted to thescattermg of alpha particles by a thin piece of foil.

In the spring of 1896, the French physicist AntoineHenri Becquerel (1852-1908) found that uranium emitsrays that act much like X rays. Just like the X rays ofRoentgen discovered several months before, the uraniumra~s fog photographic plates and pass through opaqueobjects. Their absorption is proportional to the densityof the object placed between the uranium and a photo­grap.hic plate. If the body is opaque to these rays, theoutlines of the object are formed on the plate. The urani­um rays, like X rays, are capable of ionizing air; theamount of ionization of the air is a good measure of theirintensity.

What is similar in the discoveries of Becquerel andRoentgen is the element of chance: they were accidental.

5. The Structure of Atomic Nuclei 143

But an accident all by itself is never the source of an im­portant scientific discovery. Just as there were peoplewho had "seen" X rays several years before Roentgen, sothere were (as it turned out following Becquerel's discov­ery) at least three people who had noticed the blackeningof a photographic plate that had happened to be nearuranium salts. But to "see" is one thing, and to payattention to and locate the actual cause of the phenome­non is quite another thing. That is precisely what Roent­gen and Becquerel did and what their predecessorsfailed to do. Hence the honour and the glory go tothem.

The path to Becquerel's discovery went through thefollowing stages. In the first tubes, the X rays fell onglass. The glass fluoresced under the action of cathoderays. And so it was natural to think that the penetratingrays accompanied fluorescence. Becquerel began by con­ducting experiments with substances that exhibit fluo­rescence under the action of the sun's rays. He was ratherquick to see that the penetrating rays emanated from avariety of minerals containing uranium. That~covery. But Becquerel was in no hurry to report whathe had found to the scientific community. The experi­ments were repeated a number of times. And it happenedthat the sun, out of spite, refused to come out of theclouds for several days. The photographic plates togetherwith the mineral specimens lay in a drawer of his labora­tory desk for a few days. On March 1, 1896, the sunfinally came out and he could resume his experiments.nut before doing so, Becquerel decided to check the qual­ity of his plates. He went into the dark room, developedone of the plates and saw the clear-cut outlines of hismineral specimens. Now there had not been any fluores­cence and so that could not have been the cause.

Becquerel repeated the "dark" experiments and wasconvinced that his minerals were the source of the pene-

Marie Sklodowska Curie (1867-1934J-outstanding woman scien­tist. In 1898, while studying the emission of uranium and thorium(the nature of the emission was not known at that time), she dis­covered in the ores of these elements certain substances that pos­sess a far greater emission capability. She then isolated poloniumand radium. Marie Curie and her husband Pierre Curie (1859-1906)introduced the term "radioactive". The discoveries of Marie Curiewere immediately taken up by Rutherford and led to the laws ofthe radioactive decay of atoms.

Photons and Nuclei 144 5. The Structure of Atomic Nuclei 145

trating radiation that develops all by itself, without anyhelp from light outside.

A careful examination of many samples suggested toBecquerel that the source of the rays is uranium. If amineral did not contain uranium, there was no penetrat­ing radiation. And to complete the proof he decided tomake a study of pure uranium. But uranium is a rareelement. Becquerel asked his friend, the French chemistHenri Moissan (1852-1907), for some. At a meeting of theFrench Academy of Sciences, Moissan told of a methodfor obtaining pure uranium, and Becquerel reported thaturanium emits rays. These papers were delivered on No­vember 23, 1896. Only fifty years separate that discoveryfrom the atomic bomb that was dropped on Hiroshima.

A year passed and ill the autumn of 1897, two youngphysicists, the Curies-a man and wife team-began theirexperiments in a cold shed. But they worked with enthu­siasm. Marie Sklodowska Curie (1867-1934) chose for thetopic of her dissertation a study of the chemical pecu­liarities of minerpl specimens that generate the penetrat­ing radiation of Becquerel.

The intense work led from one discovery to another.First of all, it was found that, in addition to uranium,thorium also exhibits these penetrat~The inten­sity of the emission was measured by the intensity of theionization current. Curie substantiated Becquerel's con­jecture that the intensity of the penetrating rays doesnot depend on the composition of the chemical compoundsinvolving uranium and thorium, and is strictly propor­tional to the number of their atoms.

Then came a setback: the uranium pitchblende oreyields four times more ionization than it should if itcontained uranium. It is just at moments like these thatthe talent of the investigator is so important. An ordinaryperson would assume that the atoms of uranium wereto blame. But Marie Curie realized that this phenomenon10-2542

Radioadive Decay

Does something happen to the atoms that are sources ofradioactive radiation? Yes, indeed. And these events arequite amazing. In 1902, 0UI' old acquaintance Sir ErnestRutherford (1871-1937) (we have already told-disregard­ing the historical sequence of events-about his discoveryof the structure of the atom in 1911) demonstrated thatas a result of radioactive radiation there occurs a transfor­mation of one kind of atom into another kind.

Rutherford expected that this supposition, thoughbased on rigorous experimental proof, would be challengedvigorously by chemists. True enough. The point is thatby proving the transformation of atoms, we encroachon the holy of holies-the indivisibility of the atom.By asserting that we can obtain lead from uranium weare accomplishing the dream of alchemists, who havemerited as much "glory" as astrologists.

But under the weight of proof, the critics quicklyretreated, and after some time the phenomenon of naturalradioactive decay of certain atoms was incontestably dem­onstrated both by chemical and physical methods. Whatis the essence of radioactive transformation?

To begin with, it turned out that the electron raysthat make up part of the radioactive radiation emanatefrom the nucleus. But if that is so, the charge on thenucleus increases by unity and the r~ve atom isconverted into an atom next in order in the periodictable of elements.

An alpha particle carries a double positive charge andhas a mass four times that of the hydrogen atom. If anucleus ejects such particles, the atom mnst be "dis­placed" to the left in the periodic table of elements withan accompanying isotopic transformation.

It is almost too trivial to say that unstable atomsare subject to radioactive disintegration.10·

Photons and Nuclei 146

could be explained in a different way. It might be thatthe pitchblende ore contains a small quantity of somehitherto unknown chemical element capable of extremelypowerful penetrating radiation.

The conjecture turned out to be true. The titanic workthat Marie Curie carried out revealed a new elementcalled polonium (in honour of Marie's home country Po­land) and then radium (meaning ray). Radium turnedout to be nearly a thousand times more active than pureuranium.

Now let us go more quickly, paying less attention tothe historical sequence of events.

Following the discovery of radium, other substanceswere found that were sources of penetrating radiation.They all received the name "radioactive".

What is this radioactive radiation?A radioactive specimen was placed in an evacuated

box and this was surrounded by a lead shield with aslit. The rays passed through the slit, fell on a photo­graphic plate, and left a trace on it. But as soon as thebox was placed between the poles of a magnet, the devel­oped plate revealed three marks. The radioactive beamhad split up into three rays. One ray was deflected asif it were a flux of negatively charged particles, a secondray constituted a flux of positively charged particles,and one ray was not deflected at all. It was apparentlysomehow related to the X rays.

By methods we have already discussed it was demon­strated that, in the general case, radioactive radiationconsists of a flux of electrons (before it was known thatthey are electrons they were called beta rays), a flux ofatomic nuclei of helium (called alpha particles), andsome kind of hard electromagnetic radiation (called gam­ma rays).

5. The Structure of Atomic: Nuclei 147

We do not know whether there were many such typesof atoms when the earth began to cool off. But we havean excellent idea of what types of unstable atoms cannow be found in nature. It turns out that they are membersof three families, the progenitors of which are the uraniumatom of mass number 238, the uranium atom of massnumber 235, and the thorium atom of mass number 232.

Figure 5.3 depicts the first family. The first transfor­mation is the transition of 238U to 234Th; this occursdue to the ejection of alpha particles. This is followed

Photons and Nuclei

Figure 5.3

ALPHA PARTICLES

f48 5. The Structure of Atomic Nuclei f49

by two beta transformations that carry thorium to prot­actinium and then protactinium to uranium, but thistime it is a uranium isotope of m~s number 234. Thenfollow five consecutive alpha transformations that carryus down to the unstable isotope of lead of mass number214. Another two zigzags and the proCeSS of disintegrationcomes to a halt: the lead isotope of mass number 206 isstable.

The destruction of each single atom is a random event.There are atoms that are "lucky" and have a long life­time, others live for a fleeting moment.

But it is impossible to guess in any specific case whenthe given atom will be converted. We cannot name thedate of demise of our cat or dog, but every animal specieshas its mean life span. And the same goes for every kindof atom: each has a very strict mean time of existence.Of course, atomic behaviour differs fundamentally fromanimal life. The life of unstable atoms, in contrast tothe mean life span of living beings, does not dependon any external conditions. Nothing at all can affectwhat is called the mean decay time. Always, the sameportion of atoms decays in unit time:

~N~=')..t

This formula holds true only for the case where the frac­tion !iN /N is small.

The quantity ').. is a co$tant for every radioactivet~ansition. Instead of emploting that constant, it is morepIctorial to characterize the':,: rate of the process by the"~alf-life", which is the tim~ during which half of anygIven quantity of radioactive material is transformed.For different radioactive elements, this time can havean enormous range. For example, the half-life of thepr?g~nitor of the 238U family is equal to 4.5 thousandmllhon years. Contrast that with the fact that half of

Ernest Rutherford (1871-1937)-the eminent British physicistand a great experimenter. In delicate and highly original experi­ments he demonstrated the essence of radioactive decay. In hisclassical experiments in the scattering of a flux of alpha particlesby a substance, he substantiated the modern theory of the structureof the atom as a system consisting of a nucleus and electrons movingabout the nucleus. Continuing his experiments involving the bom­bardment of different targets with nuclei, he was the first to achievean artificial transmutation of clements.

Photons and Nuclei 150 5. The Structure of Atomic Nuclei 151

the atoms of the lead isotope of mass number 214 decaysin one millionth of a second.

Nuclear Reactions andthe Discovery of the Neutron

Radioactive transformations are quite similar to thechemical reaction of disintegration. Take a chemical sub­stance, subject it to heat or light, and we obtain twosubstances. Say, carbonic acid breaks up into water andcarhon dioxide, which is like what we have just consid­ered: a thorium nucleus of mass number 230 decays intoa radium nucleus and a helium nucleus.

If nuclear disintegration is possible, then there mustbe nuclear reactions that take place via the principle

A+B......,.C+D

To obtain a chemical reaction of this kind, we have tomix the molecules of substances A and B. To achieve anuclear reaction, we have to bring into collision twoatomic nuclei.

Those were the experiments that Ernest Rutherford(1871-1937) began in 1919. This was before the time ofparticle accelerators, and nuclear reactions were achieve~

by means of bombarding a substance with alpha partl:cles. After powerful fluxes of protons and other nucl81were ohtained, new nuclear reactions were discovered.It became clear that in principle an isotope of any chemi­cal element could be converted into another isotope.Even gold could be obtained from other substances. Thedream of alchemists had become reality. The first nuclearreaction discovered was of the type A+B......,.C+D, it wasthe conversion of nitrogen and helium into oxygen andhydrogen. We can write down this reaction as follows:

~.1*N + ~He......,.1~0+1H

Photons and Nuclei 152

Note that the sums of the superscripts and the sums ofthe subscripts remain unchanged. The subscripts indicatethe charge on the nucleus, the superscripts the massrounded off to a whole number, that is, the mass numbers.Thus, the law of conservation of electric charge is main­tained rigorously. As we shall see later on, the law ofconservation of mass is only approximate. And, finally,the sum of the mass numbers is preserved just as strictlyas the charge.

As early as 1920, Rutherford suspected that there mustbe a particle devoid of electric charge and close to theproton in mass. Otherwise, he said, it is difficult tounderstand how a positively charged alpha particle canpenetrate into a positively charged nucleus, since likecharged particles repel.

The particle without a charge-the neutron-was dis­covered in 1932. It is easy to see why its discovery was"held up". The point is we see charged particles throughthe tracks they leave in a gas or a photographic emulsiondue to ionization of the molecules they encounter intheir path. But an electrically neutral particle does notinteract with electrons and therefore it does not leaveany tracks. We can judge the existence of neutrons onlyon the basis of secondary effects.

The neutron was discovered in the bombardment ofberyllium with alpha particles. This reaction can be writ­ten down thus:

:Be + ~u-? I~C + &nThe symbol n stands for neutron. But how can we besure of the existence of a particle that does not itselfleave any traces? On the basis of its actions. Imaginean invisible billiard ball on a billiard table. A visibleball is rolling over the green cloth and suddenly, inexplica­bly, bounds off to the side. The physicist cannot sus­pect the laws of conservation of energy and momentum

I

!5. The Structure of Atomic Nuclei 153

to have gone wrong. And so he concludes that the visibleball collided with an invisible ball. What is more, byapplying the conservation laws he is able to determineall the characteristics of the invisible ball; this he doesby computing the angle of deviation from the line offlight and the change in velocity of the visible ball.

The number of neutrons is calculated in the followingway. A substance containing boron atoms is placed in thepath of a neutron beam. When a neutron encounters aboron nucleus it ceases to exist. The reaction that occursis the following:

lOB I 7L · 45 + on -? 3 1 + ZU

The neutron vanished but an alpha particle came intobeing. By recording these charged particles that leavevisible tracks in a variety of devices, we can make exactmeasurements of the intensity of a neutron beam.

There are many other methods for determining, withcomplete assurance, all the parameters that characterizea neutron and, generally, any electrically neutral parti­cle. An assemblage of precisely matched indirect findingsis sometimes no less convincing than actually seeingvisible traces.

Properties of Atomic Nuclei

Prior to the discovery of the neutron, physicists pic­tured the atomic nucleus as consisting of electrons andprotons. This involved many contradictions, and at­tempts to create a theory of nuclear structure were allfailures. As soon as the neutron was found in nuclearcollisions, the idea immediately arose that the atomicnucleus consists of neutrons and protons. This hypothesiswas first advanced by the Soviet physicist D. D. Iva-nenko. cleafh

It was /' . from the vory start that the neulron mass

Photons and Nuclei 154

was very close, if not equal, to the proton mass. Thisimmediately gave rise to a clear-cut interpretation of thedifferences between isotopes of one and the same element.

As we see, to each isotope we can ascribe two numbers.One is the ordinal number in the periodic table of ele­ments, Z, which is equal to the number of protons in thenucleus. The ordinal (or atomic) number determines thenumber of electrons associated with the nucleus. Now ifthat is so, then it is clear that the atomic number mustbe responsible for the chemical behaviour of the elements(this is because chemical reactions do not involve nuclei).

Now the mass number is equal to the total number ofneutrons and protons, and so isotopes of one and thesame element differ as to the number of neutrons in thenucleus.

Very precise experiments have elicited the character­istics of both particles that mlJke up atomic nuclei. Themass of the proton is equal to 1.6726 X 10-24 gram, whichis 18:36 times more massive than the electron. The protonhas a spin of 1/2 aud a magnetic moment equal to 1.41 XX 10-23 unit in the centimetre-gram-second system. Themass of the neutron is slightly greater than the mass ofthe proton, namely, 1.6749 X 10-24 gram. The neutron hasspin 112. The magnetic moment of the neutron is antipar­allel to the spin and is equal to 0.966 X 10-23 unit in thecentimetre-gram-second system.

The spins and magnetic moments of atomic nuclei arestudied by a variety of methods: use is made of opticalspectroscopy, radiospectroscopy, studies of the deflectionof beams of particles in a nonhomogeneous magneticfield. The general principles of these methods were dis­cussed in the thinl book of this series and in the precedingchapters of this book. Here we will confine ourselvesmerely to a presentation of the basic facts obtained inthe past few decades by large band of physicists.

First of all, I would like to stress that the laws of

5. The Structure of Atomic Nuclei

quantum physics pertaining to the moment of momentum(or angular momentum) hold true for all particles. Andso for atomic nuclei we can write the formula for theangular momentum as:

Here, the quantity h is the Planck constant that weetlcounter in all formulas of quantum physics.

Usually the spin is the parameter S and not this ex­pression. Theory states rigorously and experiment demon­strates brilliantly that the spin of every particle has tobe equal to 0, 112, 1, 3/2, and so forth.

Looking through the tables of values of spin of differentatomic nuclei (obtained in a variety of experiments), wefind a number of interesting regularities. First of all,in nuclei with an even number of protons and an evennumber of neutrons, the spin of the nucleus is equal tozero (He, 12C, 160). The number of nucleons (that is,nuclear particles) that is a multiple of four apparentlyplays a very big role. In many cases (though not in all)the spin of the atomic nucleus may be obtained as follows:drop the number closest to the mass number A that is amultiple of four and multiply the difference by 1/2. Forexample: in lithium-6 the spin is 2x1/2=1; it is 3/2 inlithium-7, 1 in boron-10, and 3/2 in boron-H.

The rule is rather obvious: nuclei with an even massnumber A have integral spin or spin zero, nuclei withodd A have spin equal to a multiple of 112.

The Pauli exclusion principle is applicable to bothprotons and neutrons in the nucleus. Two identical parti­cles can reside in one energy level only if the spins areantiparallel. Since the proton and the neutron are differ­ent particles, one level can accommodate two protonsand two neutrons. In this compact group with spin zeroWe pel'c()i~ the helium atom (the alpha particle).

\/

Figure 5.4

.. ,;-......h,.----;------""~---+--=------

157

HIGH LOWTEMPERATURE TEMPERATURE

~ ~"""c-----Ht----

~ ~~;======t:;t=======II. W 1-,J!:-----HIt----

that can only be demonstrated with the aid of v~ry in­volved mathematical equations. It turns out that III cer­tain cases bosons can, in other cases cannot, congregateon a single energy level in large numbers. If they can,we say a Bose-Einstein condensation has taken place.

When a large number of particles congregate ?n onelevel their motion becomes ideally matched. TWlll par­ticle; move about identically despite the thermal chaos.

In the second book of this series we spoke of a marvel­ous liquid that possesses superfluidity at l~w tempera­tures. This liquid is a collection of 4He (helmm) atoms.

5. The Structure of Atomic Nuclei

Bosons and Fermions

We have emphasized time and again that one energylevel can accommooate only two particles with oppositespins. The time has COIlle to say that this principle (thePauli exclusion principle) holds true only for one classof particles. They are called fermions. The fermions in­clude electrons, protons, and neutrons, and also all otherparticles that consist of an odd number of fermions.There is a second class of particles called bosons. Thebosons include the photon, a number of short-lived ele­mentary particles (like, say, the pi-meson) and (most im­portant of all) all particles that consist of an even numberof fermions.

There is no limit to the number of bosons that canoccupy one energy level. To get a better grasp of thedifference between bosons and fermions, take a look atFigure 5.4. Here, each black dot symbolizes a pair ofparticles with opposite spins. At very low temperatures,bosons mainly congregate on the lowest possible energylevel. In this drawing, fermions appear as a column.

It is quite obvious that the differences in the behaviourof fermions and bosons are most obvious at low tempera­tures. At the very lowest temperatures, the number ofbosons located in the "cellar" may be almost equal tothe total number of bosons.

Don't try to "understand" what we have just said.Jlist remember it, because what we have said is reallythe ultimate truth. But I am very sorry every time I haveto tell the reader (without offering proof) about things

Photons and Nuclei 156

The presence of spin means the presence of a magneticmoment. As we know, there is a relationship of directproportionality between the mechanical mo~entum Land the magnetic moment M. Here, the magnetIc momentmay be either parallel or antiparallel to the spin.

at what temperatures? Unfortunately, theory could notpredict the transition temperature of 3He to the super­fluid state. Fantastic persistence and the overcoming ofenormous difficulties were needed before superfluid (3He)helium was obtained in 1974.

And at what temperature does the transition occur?It would be most appropriate to print the answer in bold­face type: at 0.0027 degree Kelvin. And if the readerthinks that the two-degree difference between that andordinary 4He helium is nothing much to speak of, hehas another thought coming. This is not at all like thedifference between 20° C and 18° C. In this everyday event,the temperature fell by a factor of 293/291, whereas inthe case we are discussing it dropped 1000 times-atremendous achievement of experimental physics and atriumph of theoretical physics that predicted the couplingof atoms of 3He into a "boson" pair.

A visual image might help in understanding and re­membering this. Take a look at Figure 5.5. The magneticmoments of two atoms are in the same direction. Thus,the transition of 3He into a state of Bose-Einstein con­densation must be accompanied by a jump-like change inthe frequency of the magnetic resonance. The point isthat the pair behaves like a single whole, which is pre-

Photons and Nuclei 158

The atoms of this isotope are bosons. At a temperatureof 2.19 degrees above absolute zero there occurs a conden­sation of the particles that imparts to the liquid anamazing property: superfluidity. The loss. of friction maybe explained in very rough fashion as follows: if onlyone atom gets through an extremely narrow slit, all theothers follow suit immediately.

Actually we are dealing with two phenomena, where aflux of particles moves without regard for obstacles. Thesuperfluid flow of 4He atoms resembles electric currentwith zero resistance which is found in many metals andalloys and also at low temperatures.

But electrons are fermions. They cannot congregatetogether. The way out of this impasse was found in 1956when three American scientists proposed a theory in ac­cordance with which electrons can form into pairs belowa certain temperature; and as we pointed out at the verystart, a pair of fermions is a boson. Consequently, super­conductivity sets in when such bosons condense on asingle energy level. These two remarkable phenomena,superconduc ti vi ty and superf!uidi ty, have one and the~a~,e explan?tion'"A single particle picks out a path thatIS more sUItable and all other particles follow it.

If the idea of the conversion of fermions into bosonsdue to coupling into pairs is true, then we may ask: Canthe i~otope of ~elium of mass number 3, which has spinand IS a fermIon, become superfluid like 4He?

It was evident from the very beginning that if thisphenomenon does exist, then at any rate at temperaturesmuch lower than the temperature of the transition of thebasic isotope of helium, 4He, into the superfluid state.The reason is clear: the nucleus of the 3He atom consistsof two protons and one neutron, which means it is 25 %ligh~er than its brother. Therefore, naturally, its thermalmotIOn will be more intense and setting up a march ofbosons will become possible at lower temperatures. But

5. The Structure of Atomic Nuclei

Figure 5.5

15()

The Mass and Energyof an Atomic Nucleus

cisely what was detected in the experiment. This wassuch a brilliant page in the history of physics that itwould have been a pity not to have told the reader aboutit despite any possibility of explaining under what con­ditions and on the basis of what events fermions coupleup into boson pairs.

It was mentioned in passing that the mass numberrounds the exact value of the mass of the nucleus to awhole number.

The accepted way today (this was discussed in thefirst book) is to choose the atomic mass unit as 1/12 ofthe mass of the carbon isotope 12C.

The relative masses of the isotopes of all atoms differfrom whole numbers very slightly but still so substan­tially that we cannot attribute these differences to exper­imental errors. The mass of 1H is equal to 1.00807,.the mass of deuterium is not at all twice as much, it isequal to 2.01463.

A careful study of the tables of isotopic masses showsthat the mass of the nuclei is less than the sum of themasses of the elementary particles that constitute thenew nucleus. For example, the mass of a neutronis 1.00888, the proton mass is 1.008807; the mass of twoneutrons and two protons is equal to 4.0339, yet the massof the nucleus of a helium atom, which consists of twoneutrons and two protons, is not equal to that number,but to 4.0038. Thus the mass of the helium nucleus isless than the sum of the masses of the component parti­cles of the nucleus by the amount of 0.0301 atomic unit,which is thousands of times greater than the measurementaccuracy.

Photons and Nuclei 1RO $. The Structure of Atomic Nuclei 161

There can be no doubt that these small differences havea profound meaning. What is it?

The answer is given by the theory of relativity. Itsappearance on the stage at that moment was undoubtedlymore effective than when experiment confirmed the de­pendence of the electron mass on the velocity of the elec­tron. The fact that the sum of the masses of the protonsand neutrons making up the nucleus is less than the massof the nucleus (this is termed the mass defect) is interpret­ed precisely and clearly with the aid of the famousequation E=mc2 • If a system acquires or loses an amountof energy AE, then the mass of that system increases(or, respectively, decreases) by the quantity

Am = t1Ec2

The mass defect of the nucleus (from the viewpoint ofthis principle) thus receives a very natural interpreta­tion: it is the measure of the binding energy of the nuclearparticles.

In chemistry and physics, the binding energy is under­stood to be the work which has to be expended in orderto disrupt the given bond completely. If it were possibleto split the nucleus up into elementary particles, thenthe mass of the system would increase by the amount ofthe mass defect Am.

A breaking up of the nucleus would lead to the releaseof a tremendous amount of energy. It is simple to calcu­late that a change in mass by one thousandth of a relativeunit, that is, by 1.66 X 10-27 gram, is equivalent to approx­imately 1 million electron volts.

Knowing the atomic mass of the nucleus, the reader willimmediately see something very interesting. If we dividethe energy binding the protons and neutrons in thenucleus by the number of particles, we get one and thesame number, namely, 8 MeV (megaelectron volts, or11-2542

,//

Photons and Nuclei 162

million electron volts) for all nuclei, with the exceptionof some of the very lightest ones. From this there mostdefinitely follows an important consequence: only theclosest lying protons and neutrons interact, which meansnuclear forces operate over short distances and becomepractically zero if one recedes from the proton or neutronto distances of the order of the dimensions of these par­ticles (that is to say, 10-13 cm).

The quantity 8 MeV may be compared with the energiesof chemical bonds of molecules. This comparison yieldsthe interesting fact that the chemical bond energy isusually equal to several electron volts per atom. Thismeans that several million times less energy is neededto break up a molecule into atoms than to disrupt (fission)the atomic nucleus.

From the foregoing it is clear that nuclear forces attainfantastic values. It is also obvious that nuclear forcesconstitute a new class of forces since they are capableof holding together particles that have like charges ofelectricity. Nuclear forces are reducible to electric forces.

The reg~laritie~ that these two kinds of force obeyare very hIghly dIsparate. Electromagnetic forces dimin­ish slowly, and instruments record electromagnetic fieldsat enormous distances from charged particles. Nuclearforces, on the contrary, fall off very rapidly with distance.Practically speaking, they do not operate beyond thelimits of the atomic nucleus.

Another important difference is that nuclear forces(very much like chemical valence forces) possess theproperty of saturation. Each nucleon (that is, proton orneutron) interacts with a limited number of its closestneighbours. There is no such limitation to the action ofelectromagnetic forces.

So there are three kinds of force in nature: gravita­tional, electromagnetic, and nuclear. Is that right? Sofar we do not know for certain. Physicists claim there

5. The Structure of Atomic Nuclei 163

is a fourth force with the rather inapt name of "weakinteraction". We will not go into that matter here, allthe more so since there is some hope that it will be re­duced to the electromagnetic forces.

The Energy of Nuclear Reactions

We have illuminated two important facts. First, atom­ic nuclei can be involved in reactions that take place inaccord with schemes that are very much like those famil­iar to chemists; second, the original nuclei and the newlygenerated particles will always differ slightly as to mass(this is because the sum of the mass numbers is preservedbut not the sum of the masses of the nuclei prior to andafter the reaction).

And besides we also saw that negligible differencesin mass are accompanied by the release or absorption oftremendous amounts of energy:

The energies that are released or absorbed during nu­clear transformations can in no way be compared withthe heat released in chemical reactions. Let us take someexamples to illustrate this point. One gram of coal isburned and releases heat sufficient to raise half a glassof water to the boiling point. Now the amount of heatgenerated in nuclear transformation is, as we said, nocomparison: if it were possible to break up all the nucleiin one gram of beryllium using alpha particles, the heatreleased would suffice to bring to the boiling point onethousand tons of water.

This fact was well known to Rutherford and his co­workers but nevertheless Rutherford said he thought theutilization of nuclear reactions for practical purposes animpossibility (at that time, physicists had not the slight­est inkling of the idea of chain reactions). We mightpoint out here that he was just as incapable of foreseeingthe coming revolution that his discovery had wrought11'

16!iPhotons and Nuclei 164

as were Michael Faraday (1791-1867) and Heinrich R u­dolf Hertz (1857-1894) concerning the interesting psycholo­gical conjecture mentioned in the third book of this serie~.But since we know what followed the modest expen­ments of Hutherford, we must take some time off toremind the reader of the mechanism of release and ab­sorption of energy in reactions.

First, let me state the similarities between chemicaland nuclear reactions.

Reactions of the type where particles A and Bareconverted into particles C and D release or absorb heatdepending on whether slow particles are turned into fastparticles or fast particles are turned into slow particles.That is the situation in chemical reactions and the samegoes for nuclear reactions. Furthermore, if fast particlesare created out of slow particles, this means the kineticenergy of the system increases. But the law of conserva­tion of energy permits this only if the potential energyof the system has diminished. That is to say, in thiscase the sum of the internal energies of particles A and Bis greater than the sum of the internal energies of parti­cles C and D. That is the situation in chemical reac­tions and the same goes for the internal energies ofnuclei.

According to the Einstein law, any reduction in theinternal energy is uniquely related with a reduction inmass. An increase in the internal energy leads to anincrease in mass. That is the situation in chemical reac­tions and the same goes for nuclear reactions.

But in chemistry the law of conservation of mass isoperative. The sum of the masses of molecules A and Bis equal to the sum of the masses of molecules C and D.Now, in nuclear reactions, that equality is not main­tained. So there is a difference! No, there is not. Thedifference is only quantitative. In chemical transforma­tions, the changes in energy (and hence in mass) are so

5. The Structure of Atomic Nuclei

slight (slight from the point of view of relativistic theory)that the changes in the masses of the molecules cannotbe detected experimentally. Thus the analogy betweenboth types of reactions is one hundred perce~t.

Because this is so important (many people thlllk thatthe release of nuclear energy is some kind of specialprocess, but that is not so), let us reason in. similar ~ash­

ion for the case where particle A decays llltO pa~tlCles

Band C. If the particle splits up into parts. all by Itself,then we say that particle A is unstable. If A IS a molecul~,

we say that the substance decays (disintegrates). If A ISa nucleus, we say the substance is radioactiv~. In bothcases heat is released. Particles Band C wIll possesssome' kind of kinetic energy that they did not possessbefore. This energy came from the po~ential ~nergy.Pictorially speaking, the spring connectlll?, p.artlCles Band C broke; to put it scientifically, the bllldlllg energyvanished. It was due to this binding energ~ that we ob­tained fast-moving particles Band C, that IS, the energywas released in the form of heat.

Because of the small amount of energy, in a chemicalreaction we do not detect any difference in the' mass ofmolecule A and in the sum of the masses of molecules Band C that originate from A. Now in the case of m~clearreactions this difference is readily detected. expenmen­tally. Nu'clei Band C will have a mass exceedlllg the massof nucleus A by the amount of the mass defect.

The very fact that a certain reaction gene:ates heatdoes not yet mean that it will have some practIcal. value.The condition of instability of a system? that IS,. thecircumstance that the original substance I.S at. a lugherenergy level than the products of the reactI?I~, IS, as themathematician would say, a necessary conditIOn but nota sufficient condition. . .

In book two we discussed at length the condItIOns t.hatmust be fulfilled for a substance to serve as a chemIcal

Photons and Nuclei 166

fuel. .Here we need merely continue the analogy betweenchemIcal and nuclear reactions.

To. sum~arize, th~n: it is not enough that a chemicalre~ctlOn ywld heat, It is necessary that the heat "ignite"adJacent molecules.

It is therefore clear that having learned to make atomicnuclei collide with the release of tremendous amountsof energy, physicists had not in the least come closeto the creation of nuclear fuel.

When alpha particles bombard beryllium or lithium,:hese ele~ents do. not behave like a fuel. They do, how­ever, satIs!y ~he fIrst requirement of a fuel: they produceenergy. LIthIUm a~d beryllium behave like tiny piecesof coal, each of whIch has to be ignited with a separatematch.

Right up to the end of the 1930s, the creation of nu­clear fuel was regarded as a totally hopeless undertaking.

5. The Structure of Atomic Nuclei•

FROMSECONDS

TO YEARS

167

A Nuclear Chain Reaction

Beginning from 1934, mainly due to the work of theItalian physic.ist Enrico Fermi (1901-1954) and his col­laborators, eVIdence was obtained that the atomic nucleiof most elements are capable of absorbing slow neutronsand thus becoming radioactive.

At that time, certain radioactive transformations werekno~n to consist in the emission of electrons and alphapartICle.s (~hese tran~formations are accompanied by gam­~a. radIat~on). But m 1938 a number of investigators (itIS mterestmg t? note that the fundamental discovery weare about to dISCUSS was not done by one person) foundthat uran ium activated by neutrons via Fermi's methodcontained an element similar to lanthanum. There couldbe only one explanation: under the action of neutronsthe uranium atom breaks up into two more or less equalparts. The exceptional importance of this discovery was

Figure 5.6

clear from the start. The point is that by that time thefollowing regularity had been discovered: the greater theatomic number of an element the more neutrons thereare in the nucleus. In uranium, the rat~o of the ~umberof neutrons to the number of protons IS approXImatelyequal to 1.6. And for elements like, say, lanthanum,located in the middle of the periodic table of elements,this ratio ranges between 1.2 and 1.4.

Now if the uranium nucleus fissions (that's the term)into two roughly equal halves, then the nuclei of thefission products will invariably contain some "extra"neutrons. They will eject neutrons, and neutrons playthe role of "matches".

The possibility of a chain reaction was now evident.

Figure 5.7 shows that for the most part the nucle~sof uranium-235 splits into two unequal frogments. As ISevident from a glance at the curve, most fissions resultin the formation of nuclei with mass numbers 141 and 95.

The set of newly produced radioactive fragments isvery great at any rate. The ~~s~ varie~ req?irements ofindustry with respect to artIficIal radIOactIve elementscan now be satisfied.

If the neutrons formed in the fission of one nucleuscan be made to fission the nuclei of other atoms of urani­11m, then a chain reaction can be set up.

Since matter is extremely "full of holes" as regardsits nuclear structure, it is highly probable that the neu­trons formed in the fission of a nucleus will leave the

Photons and Nuclei 168

The first calculations of this phenomenon were carriedout in 1939. The dramatic sequence of events, frcim thefirst ato~ic pile (now called reactor) to the making ofthe atomIC bomb and its explosion at Hiroshimp hasbeen discussed in all its detail in dozens of books' butthat is beyond the scope of our story and we will co'nfineourselves to the present-day state of the matter.

We have three ~uestion~ to take up: first, the meaningof a nuclear cham reactIOn, second, how the reactioncan be harnessed, and, third, when it results in an ex-plosion. ,

~igure 5.~ is a dillgram of one of the most importllnt re­actIOns of thIS type: the fission of the uranium-235 nl;l.cleus.

The ~uestion of the first neutron is simple: it ~an befound m the atmosph~re. If' we want a more 'active'~match", we can take a tiny mixture of radium and beryl­hum.

A neutron enters the nucleus of a uranium-235 i atom~hich c?nsists of 92 protons and 143 neutrons packedtightly mto a sphere of radius of about 1042 cm andproduces an isotope called uranium-236. The visitor de­forms the nucleus and, after a time interval of about10-14 second, the two halves of the nucleus are held to­gether by a tiny bridge; in another 10-14 second the nu­cleus has split into two pieces. At the same time each ofthe fragments ejects two or three (an average 'of 2.56)neutrons. The fragments fly apart with a colossal kineticenergy. One gram of uranium-235 produces as muchenergy as 2.5 tons of co~l, or 22 000 kilowatt-hours.Then, 10-12 second later the nuclei formed in the fissionprocess have more or less calmed down with the emissionof eight photons of gamma radiation. The new nucleiare radioactive. Depending on the kind of fragments thath.ave formed, the subsequent process of decay may con­tmue for seconds or for many years with the emissionof gamma rays and the ejection of electrons.

5. The Structure of Atomic Nuclei

6

5

~4III

~ 3iii~2

Figure 5.7

169

171Photons and. Nuclei 170

substance without fissioning any other nuclei. What ismore, we must not forget that not every encounter be­tween a neutron and a nucleus leads to fission. A chainreaction is ensured only if at each instant of time thenumber of neutrons located inside a piece of substanceis the same or greater than the number at the precedinginstant. This condition is stated by the physicist asfollows: the neutron multiplication factor-which is equalto the product of the number of neutrons into the proba­bility of a neutron encountering a nucleus and into theprobability of neutron capture by the nucleus-must notbe less than unity.

That is why pure atomic fuel has a critical mass. Ifthis mass is less than critical, one can calmly (well,more or less calmly, let us say) carry a piece of nuclearfuel in one's pocket. And it won't be heavy because thecritical mass is close to one kilogram.

Naturally, it is very important to know the figl~refor the critical mass. The first calculations of this quantitywere carried out in 1939 by Francis Henri Perrin (1901~ ),the son of Jean Baptiste Perrin (1870-1942). This cal­culation is only of historical interest today hecause atthat time it was not known that a chain react ion cannotdevelop in natural uranium, no matter how much of itis taken. But not much time was needed for the pictureto become quite clear. A chain reaction does not developin na turl'll uranium because the neutrons produced inthe fission of the nuclei of uranium-235 are absorbed viawhat is called "resonance" capture by the atoms of urani­um-238 with the formation of uranium-239, which, bymeans of two successive beta disintegrations, turns intoneptunium and plutonium. Only uranium-235 and plu­tonium possess a critical mass. Substances with a criticalmass constitute nuclear fuel. These were the facts thatphysicists had at their di~posal at the beginning of the1940s.

5. The Structure of Atomic Nuclei

If a device could be designed in which pressing a but~onbrought together two pieces of nuclear f?e.l, each of whIchhas a mass less than critical and, when Jomed, ~as a ma~Jexceeding the critical mass, t~en. an explOSiOn .wontake place. That is th? simple prmciple that underhes theworking of the atomIC bomb. f

Now what do we have to do to control the course 0 anuclear reaction? Quite obviously we have ~o reft~ ~system containing not only the atoms of t e ne ualso other atoms capable of absorbing neu~rons a~d, asit were, putting them out of action. ~adIU:mm ro s ar~quite suitable for this purpose. Cadmmm IS a ~o;.d abfsorber of neutrons. If we make a structure conSlS mg 0

rods of nuclear fuel and rods of cadmium and then arran~efor inserting and extracting the rods. (thi~ takes plflc~mthe body of a nuclear reactor-at fIrst It was ca e. a

'1 ) can set up a nuclear chain reaction by. maklll~i~ee~e::ron multiplication factor slightly modre ~h~n1UUli

. hen we can raise the heat release to the eSlre eve;~'d\nsert cadmium rods so that the multiplication factorbecomes exactly unity.

6. Energy Around Us

Sources of Energy

In the final analysis, it will be seen that all the energygenerated here on the earth comes from the sun. Tem­perat~res ?f the order of millions of degrees rage in thedeep mterIor of the sun. That is where reactions developbetween atomic nuclei. They go by the name uncontrolledthermonuclear fusion.

Man has harnessed these reactions and on the earththey release fabulous amounts of energy. The hydrogen~omb, for ~xample. Research scientists are now attempt­mg to brmg the thermonuclear fusion reaction undercontrol. That is the topic of this chapter.

Now that we have discussed the structure of nucleiwe are ready to ~xamine .t~e sources of terrestrial energy.

Under terrestnal condItIOns, we can obtain energy in~hree ways. First, we can extract energy from fuel (chem­I~al ~r nuc~ear). This requires setting up a chain reac­tIOn m wInch molecules or atomic nuclei are combinedor disrupted. Chemical fuels of practical value includecoal, petroleum, and natural gas. Nuclear fuel for fission(breaking up of particles) includes uranium and thoriuma~d nuclea.r fu~l for fusion (combining of particles) con~stltutes prImarIly the light element of hydrogen .

.Second, the conversion of kinetic energy into work.Illvers can be made to generate electricity and thus"work". Hydraulic power ("white coal") becomes one ofthe most important sources of energy. Here water falling

6. Energy Around Us 173

from considerable heights generates electricity; this canbe achieved by constructing dams or by utilizing naturalwaterfalls, like the Niagara Falls. Similarly, wind canbe harnessed to convert energy into work turning' theblades of a windmill. We shall see that wind ("bluecoal") must be taken seriously. Windmills are coming tolife again on a much higher engineering level and willsoon be making a substantial contribution to the energybank. The same can be said of such an energy source astidal waves.

The laws of thermodynamics suggest a third solutionto the energy pro~lem. In principle, it is always possibleto design an engine that can use temperature differences.As we know, a transfer of heat from a hot body to a coolbody can be partially converted into mechanical work.Now we encounter temperature differences in manyplaces: in the earth's crust, in the oceans, and in the atmosphere. Going down into the interior of the earth (actually,of course, it amounts to digging slightly into the crust)we invariably observe an increase in temperature.

To repeat, all these three possibilities ultimately stemfrom the fact of solar radiation coming to the earth.The earth receives an insignificant portion of the energyradiated by solar rays. But even this minute bit is socolossal that for many years it will handle our needsand be enough to carry out the most fantastic projects.

But solar energy can also be utilized directly. Thereader will recall that we have learned to convert radiantenergy into electric current with the aid of photoelectricdevices. As we know, the generation of electricity is themost important way of making energy serve practical­ly.

Of course, in many cases both the internal energy ofmatter and the energy of the motion of water and windcan be utilized directly, bypassing the stage of conversioninto electric current. However, perhaps in all cases, with

175Photons and Nuclei 174

the exception of aircraft and rockets, it is advisable toobtain electricity from the prime source at power plantsand make the electric current serve our needs. The im­portance of electric energy will grow immeasurably assoon .as we learn how ~o construct light-weight high­capacity storage battenes for electric automobiles inplace of the heavy low-capacity ones now in use.

Before going on to a concrete discussion of energysources, I wish to bring the attention of the reader oncemore to two important classifications of sources. Firstof all, there is an important dividing line between fueland solar energy. In the former case. we are spendingthe wealth of the earth which cannot be replaced. Nowthe sun, the Wind, and the water are sources for whichyve do not have to pay anything. They are free of chargeIII the. s~n~e that using their energies does not in anyway dlmlmsh the wealth of the earth. Windmills do notreduce the amount of air in the atmosphere, the operationof hydropower plants does not take water out of therivers, and solar-dri~en power-generating devices do notuse up any terrestnal resources.

There is another classification. We have to protect?ature at large-the flora and fauna of the earth. ThisIS a ~roblem .whose importance is hard to overestimate.Burmng fuel IS bad both because it depletes the materialof the earth. and. also because it pollutes the ground,water, and aIr with enormous quantities of deleteriouswaste. Even hydroelectric power plants have their draw­backs. Changes made in the water regimen of rivers canaffect the climate and complicate the life cycles of thefish population of the world.

There can be no doubt that the optimal mode of obtain­ing energy is the direct utilization of the sun's radiation.

Such are the principles. Now let us take up a morec?ncrete discussion of the possibilities of energy utiliza­tIOn under terrestrial conditions and give some of the

6. Energy Around Usfigures that can be found in the reference literature onpower generation.

We start with solar energy. Every square metre on theouter boundary of the earth's atmosphere receives about1.4 kW of energy (which is about 10 million kilocaloriesof energy in one year). Hundreds of kilograms of coalwould be needed to generate that much heat. So howmuch heat does the earth as a whole receive from thesun? Calculating the area of the earth and taking intoaccount the nonuniform illumination of the earth's sur­face by the sun's rays, we obtain about 1014 kW. This is100 000 times more energy than that produced by all thepower sources on the earth: factories, power plants, aut~­

mobile engines, aircraft engines; in other words, thiSis 100 000 times more power than the energy consumedby the entire population of the world (about a thousandmillion kilowatts).

Up to now, solar energy has hardly at. all b~en touched.The point is this: although the overall figure IS enormous,a good deal of the energy falls on the slopes of inacces­sible mountains, on the oceans that occupy most of theearth's surface, and on the sands of desert areas. Whatis more only a relatively small amount of energy falls onany sm~ll area. Surely it would not be feasi.ble to con­struct energy receivers over many square lulometre.s ofsurface. And, finally, solar energy can be converted Illtoheat only in localities where there are regularly manysunny days during the year.

The thirst for power and tremendous advances in theproduction of semiconductor photoelectric c~lls have rad­ically changed the psychology of .power eng~neers. Thereare many designs and even ex~eI'lmental umts that ,focussolar radiation on thousands (Ill the future there wIll bemillions, perhaps eVlm thousands of milli~ns) of photo­electric cells. And cloudy days and absorptIOn of rays bythe atmosphere are not serious deterrents. There can be

Photons and NucleI 176

no doubt that there is a great future in the direct utiliza­tion of solar energy.

Wind is now regarded in a different light too. Sometwenty years ago wind was dismissed as a reasonablesource of power. It has the same defect as solar energy,and that IS that the amount of energy per unit area isrelatively small. In order to generate power for factories,one would need windmills with vanes too large to be prac­ticable.

Another essential drawback is that the wind forceis not a constant factor. All that relegates the energyof the wind, or "blue coal" as the poet says, to smallengines (Windmills). When the wind is up, these powerplants generate electricity for agricultural machines andhome lighting. If extra power is generated, it can bestored in storage batteries, which will take up the load intimes of calm. Obviously, the windmill has only a sec­ondary role to play.

Today, engineers discussing the energy problem reasondifferently. Designs for electric power plants consistingof thousands of appropriately positioned "Windmills" withenormous vanes are on the verge of being made intoh.ardware. T~e ~se of wind ;-rill surely make an appre­CIable contrIbutIOn to man s ever increasing needs.

Another gratuitous source of energy is moving waterthe tidal waves of the ocean that constantly wash up ont~the land, and the water of the rivers that flow to theseas and oceans. In 1969, hydroelectric power generationcame to 115.2 thousand million kilowatt-hours in theUSSR and 253.3 thousand million kilowatt-hours in theUSA; however, utilization of water resources in the So­viet Union is only 10.5 % while in the United States itcomes to 37 %.

These figures are impressive, but' if we were deprivedof all coal, petroleum, and other energy sources and onlymade use of hydro power-even if all engineering oppar-

6. Energy Around Us 177

tunities were utilized for hydroelectric power g~neration­we could not meet our requirements today.

What energy can be extracted from tidal waves? A greatdeal, although it is approximately one tenth the energyof rivers. Unfortunately, only a small portion of thisenergy can be harnessed. Because of the pulsating natureof the tides, there are many complications. However,Soviet and French engineers have found some practicalways of overcoming them. Today, tidal power stationsensure guaranteed outputs during hours of maximum con­sumption. In France, a tidal power station has been con­structed on the river Rance, in the USSR there is a stationat Kislaya Guba in the vicinity of Murmansk. This latterstation is an experimental model for tidal power stationsof 10 000 million watts (now in the design stage) to bebuilt in bays of the White Sea.

Ocean water at considerable depths has temperatures10° to 20° different from the surface layers. This offerspossibilities of constructing a heat engine whose heater,in the medium latitudes, would be the upper layer ofwater and whose cooler would be the deep-lying layer.Such machines could have an efficiency of 1 % to 2 %.This of course is also a highly unconcentrated source ofpower.

Another gratuitous source of power is geothermal ener­gy. We will not discuss countries that have many geysers.There are not so many; and where there are, the heatof the geysers is utilized for industrial purposes. Whatwe shouldn't forget is that two or three kilometres belowany spot on the globe we encounter temperatures of theorder of 150 to 200 degrees Celsius. The principle un­derlying a geothermal power station is obvious. Drilltwo channels. Cold water enters one and hot water comesout of the other.

12-2542

Fuel

All the power sources described abo:~ ~ave great ad­vantages over fuel. Fuel is burned .. UtIhzmg the en~rgy

of coal, petroleum, and wood constItutes largely an ure-placeable destruction of terrestrial wealth. .

What are the reserves of fuel here on earth? Ordmaryfuel that burns when ignited with a match includes coal,petroleum, and natural gas. Their ~eserves are extremelylimited. At the rate we are now usmg up petroleum, theknown reserves will have .been depleted completely bythe beginning of the next century, the same goes for g.as.There are somewhat larger reserves of coal,. somethmglike ten million million tons. The combustIOn of onekilogram of coal yields 7000 kilocal?ries o~ ~eat: (Thereare all kinds of fuel, and quality vanes. ThIs ~s ~aturall.yonly a standard fuel used for measurem.ent; It IS a umtof comparison fuel used to compare dIfferent types ofmaterial.) Thus, the overall energY2~up:pliesof.coal co~eto something on tIle order of 10 kII?ca]ones, whIchis roughly a thousand times the world s annual powerconsumption.

This thousand-year reserve is actually not much. A

Photons and Nuclei

-n ................. •. --:__

Figure 6.1

178 6. Energy Around Us 179

thousand years is a great deal if compared with theaverage human life span, but a person's life is only amoment when compared with the time of life on earthand with the existence of human civilization. What ismore, power consumption is constantly increasing perhead of the population. For this reason, if the fuel sup­plies consisted solely of petroleum and coal, the energy­reserve situation here on earth would have to be regardedas catastrophic.

But must we confine chemical fuel to the naturallyfound substances? Of course not. In a number of cases,it may prove that synthetic gaseous or liquid fuel isbetter than petroleum and natural gas.

In recent years much attention has been paid to theindustrial production of hydrogen. As a fuel, hydrogenhas many good points. It can be produced in limitlessquantities in a variety of ways. It is readily availableeverywhere and so there are no transportation problems.Hydrogen can easily be purified of any undesirable im­purities. In a number of cases it will be found thatburning hydrogen directly for heat is the best thing.We can bypass the stage of converting it into elec­tricity.

At present, there are three main profitable ways ofobtaining hydrogen: electrolytic, thermochemical de­composition, and, finally, irradiation of hydrogen-con­taining compounds with neutrons, ultraviolet rays, etc.It is even economically profitable to obtain hydrogenfrom coal and petroleum in nuclear reactors. In thesecases, the hydrogen can be delivered by pipeline to thesite of consumption, as is done with natural gas.

Concluding this brief survey of chemical fuels, let usnow ask the question: What is the situation like concern­ing nuclear fuel? What are the reserves of nuclear fuelon the earth? Remember that only small amounts areconsumed in nuclear reactors. One kilogram of nuclear12·

Photons and Nuclei f80

fuel produces 2.5 million times more power than a kilo­gram of coal.

Approximate estimates show that the reserves of poten­tial nuclear fuel (why the word potential is used will beclear from what follows) come to about 2 million tons ofuranium and 4 million tons of thorium. These are the fuelsused to generate power in nuclear reactors based on thenuclear fission process. Will any other materials be addedto these two elements? Perhaps. The number of nuclearreactions that produce energy is tremendous. The wholeproblem is how to make the process a chain reaction.

Let us first see what we are able to do at the presenttime. As follows from the preceding chapter, there isonly one naturally found substance that can be used asnuclear fuel. That is the isotope uranium-235. The ura­nium mined in the earth contains 99.3% of uranium-238and only 0.7% of uranium-235.

At first glance the simplest idea is this: isolate thenecessary isotope, build reactors consisting of pieces (orrods) of that substance, introduce into the body of thereactor control rods that absorb neutrons and, thus, con­trol the nuclear reaction.

First of all, note that by absorbing neutrons and notallowing them to participate in the chain reaction wereduce the power of the reactor; in other words, we extractsmaller amounts of energy per second from a unit mass ofnuclear fuel. Now, if we slow down neutrons to thermalvelocities and convert the "fast" neutrons generated inthe disintegration of a nucleus into "slow" neutrons, thatwill be extremely useful in enhancing the efficiency ofthe reactor. This is because the nuclei of uranium-235absorb slow neutrons with a far greater probability.

In all designs (except experimental reactors thathaven't gone beyond the laboratory stage), the moderatorhas been either heavy water or ordinary water. Heavywater is good because it does not absorb neutrons at all,

6. Energy Around Us 181

but it is not so good at slowing down neutrons as ordinarywater is.

Thus, the best and simplest method is to isolate theuranium-235 isotope. That, as you recall, is a very expen­sive procedure. Chemical methods are useless: after all,these are chemically identical substances.

At present the most profitable method is by centrifug­ing. Before this operation can be started, we have toobtain a gaseous compound of uranium. The only suchcompound that is in the gaseous state at ordinary temper­atures is uranium hexafluoride. The mass difference ofthe gas molecules containing the isotopes uranium-238and uranium-235 is so slight that the best centrifugeenriches the gas with lighter molecules only up to 12 %.In order to obtain uranium with 3 % of uranium-235(such fuel can then be used in a nuclear reactor), the pro­cess has to be repeated 13 times. Clearly, the true solutionof the engineering problem cannot be the obtaining ofpure uranium-235.

But there is another and perhaps more important rea­son. Without uranium-235 it is impossible to convert thebasic mass of uranium (and also thorium) into nuclearfuel. That is precisely why we spoke of potential fuel.As for the isotope uranium-235 itself, that fuel shouldhold off the onset of an energy famine for a hundred yearsor so. Therefore, if we want to utilize nuclear fuel formany long centuries, the approach must be quite differ-ent.

Nuclear fuel can be produced in the reactor itself,it turns out. We can first of all make plutonium-239,which is obtained from uranium-238, and, secondly, ura­nium-233, which is obtained from thorium-232. But thereis no way of starting without uranium-235.

Reactors that generate energy and at the same timecreate new fuel are called breeders. It is possible to achievea situation where the reactor will produce more new

Electric Power Plants

Of course, there are many instances of the direct util­ization of energy not in any way connected with thegeneration of electricity. Gas burning in the kitchen, thelaunching of a space vehicle (here, the recoil of theproducts of combustion sends the rocket into space), andeven the old steam engines that are occasionally still

183

wind is converted directly6. Energy Around Us

found. In a number of cases,into motion and energy.

But in most cases we need electric current-to providelighting, to power electric engines, to provide for electrictraction, to ensure the operation of electric-welding andheating furnaces, to charge storage batteries, and to domany other things. At any rate, today we cannot con­ceive of delivering energy over distances in any way otherthan by electric current. It is therefore no exaggerationto say that the principal hero of technology remains theelectric power plant.

There are still two basic industrial methods of turningthe rotating parts of electric engines, the machines thatgenerate electricity. If this work is done by the energyof falling water, we speak of hydroelectric power plants;if the moving force is the pressure of steam on the bladesof turbines, we speak of thermal power plants.

Included in the class of thermal plants are atomicpower plants, although they actually differ from ordinarythermal plants in only one way: they use a different typeof fuel. But in both cases we obtain heat that is used toproduce steam.

The urbanite today is acquainted with the term centralheating and power plant. These plants combine a supplyof electricity with steam and hot water.

The energy of falling water has been utilized for cen­turies. The water wheel of the ancient mill is the proto­type of the modern hydraulic turbine. A stream of waterhits the blades of the wheel and transfers part of itskinetic energy. The blade moves and rotates the wheel.

It is not so easy to position the blades of a wheel inorder to obtain the best efficiency. This engineeringproblem is resolved by specialists in different ways, de­peulling on how the water falls. Naturally, the greaterthe height (sometimes even 300 metres has been attained)from which the water falls the better the turbine performs.

182

that is, the reproduction factor

Photons and Nuclei

fuel than it consumes,is greater than unity.

To summarize then: technologically feasible ways ofutilizing all the supplies of uranium and thorium areavailable. Consequently, by the most conservative esti­mates, there is enough fuel to last thousands of years.

And yet adding uranium and thorium to thB fuel listdoes not fundamentally solve the energy problem formankind; after all, there is a limit to the reserves ofmineral wealth in the earth.

Now thermonuclear reactions are quite a different prop­osition. If we are able to tame the fusion of light nuclei,achieve a controlled self-sustaining thermonuclear reac­tion, then we can truly say that the energy problem hasbeen resolved. How realistic is such a solution? Just recent­ly, physicists have obtained hydrogen plasma with temper­atures of about 60 million degrees. That is the temperatureat which a thermonuclear reaction can take place. Butthe problem is to make such a reaction self-sustainingand to design a fusion (or thermonuclear) reactor. Thatis what we have yet to do.

There is so much thermonuclear energy in the watersof the oceans that it could satisfy all the power require­ments of humanity for a time span exceeding the age ofthe solar system-a limitless source of energy, indeed.

We thus conclude our talk on fuel. Now let us examinethe machines that make the fuel work.

"The latest word in hydraulic-turbine constructipn areturbines exceeding 500 OOO-kilowatt capacity. Since pow­er outputs of this kind are generated at rather low ro­tation speeds (of the order of 100 a minute), the newhydraulic turbines are colossal both in size and weight.

Depending on the direction of flow in the rotor, hy­draulic turbines are divided into axial-flow turbines andradial-axial turbines. The Soviet Union has hydroturbinesof the radial-axial class with power outputs of 508 mega­watts and 7.5-metre rotors.

Hydroelectric power plants generate the cheapest elec­tricity, but they are several times more expensive to con­struct than thermal power plants and the constructionperiod is much longer. In hydraulic power plants, hydro­turbines set in motion hydrogenerators. These are verylarge synchronous machines mostly with vertical shafts.In such a machine, the diameter of the rotor is 7 to 10times its length and in some of the largest machines itexceeds 15 metres. This is necessary for the machine tooperate stably for variable speeds of the hydroturbinethat rotates it. The rotor of a hydrogenerator has a largenumber of salient poles. For example, the generators ofthe Dnieper hydropower station have 72 poles. .A specialdirect-current generator (called an exciter) is used tofeed the windings of the poles. Hydrogenerators operateat rather low speeds-from 80 to 250 revolutions perminute.

The hydrogenerator of the Krasnoyarsk power station(500 megawatts) operates at 93.8 revolutions per minute,has a 16-metre-diameter rotor and weighs 1640 tons. A650-megawatt generator is being designed for the Sayano­Shushensky hydropower station.

As I have already pointed out, hydropower generationaffects the environment in certain adverse ways. Butnevertheless there can be no doubt that hydropower plantshave definite advantages over thermal power plants. First

Photons and Nuclei 184 6. Energy Around Us 185

of all, hydropower plants do not use up fuel, and weknow that fuel is in short supply. Thermal power plantshave yet another very great drawback. A considerableportion of the power generated in converting the energyof the fuel into electricity is unavoidably lost.

Still and all, something like four fifths of all the elec­tric power is generated at thermal power plants by meansof turbogenerators in which the driving force is the pres­sure of steam.

To increase the efficiency of the generator, it is nec­essary to raise the temperature of the steam. This canobviously be attained only by simultaneously increasingthe pressure. In modern thermal electric power plantswith capacities ranging from 200 to 300 megawatts, theturbines operate at steam parameters of 565°C and 24meganewtons per square metre.

But why are high temperatures needed? To put it sim­ply, a steam turbine utilizes the same phenomenon thatmakes the cover of the tea kettle jump up when the waterbegins to boil. In other words, in a steam turbine wesee thermal energy transformed into mechanical energy,and then from mechanical energy to electricity. Now,in the first transformation (this can be proved rigorously),energy is lost to the extent of a portion equal to the ratioof the ambient temperature to the temperature of thesteam.

It is definitely too bad that we have to go through the"thermal stage" in modern devices in order to extractenergy. This transition always involves a very great lossof power, and the ideal power plant of the future is onein which energy of any origin can be converted directlyinto electricity. Since that most important problem isnot yet solved, there is only one thing left; and thatis to strive for the highest possible temperatures of steam,gas, or plasma.

Difficult /;\S that is, we are still able to achieve effi-

Photons and Nuclei 186

ciencies of about 40 % in thermal electric power plants.The steam power generator is an electric machine with ahorizontal shaft. The rotor is manufactured together withthe ends of the shaft as a single piece of forging madeout of special turbo-rotor steel. This is done becausethe mechanical stresses due to the high speeds of 3000revolutions per minute reach the very limits of modernmaterials. For that same reason, the rotor does not haveany salient poles. Parts of its cylindrical surface haveslots into which the field (excitation) winding is laid.A three-phase alternating-current winding is laid in theslots of the stator.

Because of enormous mechanical stresses, the diameterof the rotor is restricted, and so to obtain sufficientpower the machine has to be stretched out in length.

The first Soviet 500-kilowatt turbogenerators were man­ufactured at the Leningrad "Elektrosila" plant in 1925.In 1964, the "Elektrosila" works turned out a turbogen­erator exactly 1000 times more powerful than the firstone-500 000 kilowatts.

The desire to obtain more power from a single machinewithout increasing the already enormous dimensions hasled to very complicated designs. For instance, to reducelosses in the windings of the stator, they are made in theform of hollow copper conductors in which water constant­ly flows. The excitation winding is cooled with hydrogenunder a pressure of about 4 atmospheres. The use ofhydrogen, which is 14 times less than the density of air,made it possible to increase the power of the turbogen­erators by 15% to 20%.

The five-year plan of 1976-Hl80 calls for the manufac­tlll'e of turbogenerators of between 1000 and 1200 thousandkilowatts for thermal and atomic power plants.

One of the most interesting power plants in the worldhas been built in the Soviet Union. It is called U-25 andfeeds about 7000 kilowatts of electricity into the electric

6. Energy Around Us 187

netwo!~. This is the world's largest plant for generatingelectrICIty by the method of magnetohydrodynamics,MHD for short. An MHD-generator has no rotating parts.

The i.dea behind the. operation of this interesting gen­erator IS extremely SImple. A flux of ions having con­siderable kinetic energy (this is, a jet of plasma) passesthrough a magnetic field across the magnetic lines offorce. The ions are acted upon by the Lorentz force. Recallthat the intensity of an induced electric field is propor­tional to the rate of ion flux aud to the maO'nitude of~agnetic induction. The electromotive force is perpen­diCular to the motion of the ions. That is the direction inwhich the current arises and closes on an external load.The electrodes that receive the current are in direct con­tact with the plasma.

Electric energy is generated due to the fall in energyof the plasma jet. An MHD-generator permits raising theefficiency of the power plant to 60% and more.

The critical factor in generating cheap power fromMHD is the magnetic field in the channel. This must bean. intense field. An ordin~ry electromagnet with a copperCOlI can produce such a field but the magnet is then un­:vieldy, complex in design, and expensive. What is more,It consumes a good deal of power itself. In this connectiondesigners have developed a new concept in designingmagnets with a superconducting winding. This kind ofmagnet can generate the necessary magnetic field of re­quired intensity and with low power consumption andinsignificant heating. Calculations have shown that theconsiderable expenses needed to obtain temperatures croseto absolute zero justify themselves.

From this brief summary we see that traditional waysof increasing power generation have not yet exhaustedthemselves. However, one can hardly expect progressonly along that road.

Aside from the fact that fuel supplies and opportunities

Photons and Nuclei 188

for utilizing hydraulic sources of energy are coming to anend, one should not lose sight of the appreciable effecton the environment of the construction of new electricpower plants. Ecologists have warned of the necessity ofa very cautious approach to interfering in the life ofrivers. Power engineers arfl reminded of the enormousquantities of ash that are ~hrown into the atmosphere inthe course of fuel combustion. In one year, the earth'satmosphere receives 150 million tons of ash and about100 million tons of sulphur. Particularly disquieting isthe increase in the atmosphere of carbon dioxide. Everyyear, 20 thousand million tons of it are added to theatmosphere. During the past 100 years, the quantity ofcarbon dioxide in the atmosphere has increased 14 %.

There are two reasons for this growth: the destructionof vegetation on the earth and, what is most important,the release into the atmosphere of this "gaseous ash" inthe combustion of ordinary fuel. This ceaseless increasecan lead to ruinous consequences, of which the worst isan increase in the temperature of the atmosphere by 1.5to 3 degrees. This might seem to be a small rise but in~urn it can lead to l\n irreversible melting of the polarIce caps. Climatologists claim that the ultimate permissi­ble increase in carbon dioxide in the earth's atmospherecannot exceed several tens percent.

Nuclear Reactors

As has already been pointed out, atomic power plantsbelong to the class of thermal electric power plants. Thedifference lies in the production of steam that is thendirected onto the blades of a turbine. A nuclear reactorcould be called a nuclear boiler because of the similarityof power generation.

A nuclear reactor is ordinarily in the form of a cylin­drical building. The walls havl;l to be very thick and

6. Energy Around Us 189made of materials that absorb neutrons and gamma radia­tion. Reactors generating something like 1000 megawattsof electricity come in a variety of sizes depending onthe fuel used, the method of moderating neutrons, and themode of heat removal. But in all cases the size is impres­sive: the height of a 5- to 10-storey building and someten or so metres in diameter.

Nuclear power engineering began to develop right afterthe Second World War. In the Soviet Union, the mostimportant nuclear investigations were headed by IgorV. Kurchatov, a marvelous scientist and an excellentorganizer.

In the Soviet Union and in other countries, a numberof designs were tested. The first stumbling block is todecide the isotopic composition of the uranium or othernuclear fuel used. The engineer is then confronted withthe form in which the fuel is supplied as a solution ofuranium salts or in solid chunks. Solid fuel elementscan come in a variety of shapes. Bars can be used butlong rods appear to be most suitable. A very essentialrole is played by the geometry of the fuel elements (thepositions they are placed in). Engineering calculationsare needed to find the most suitable positions of the con­trol rods that absorb neutrons. Their movements (all auto­matic, naturally) must ensure the required value of theneutron multiplication factor.

Differences in the behaviour of slow (thermal) neu­trons and fast neutrons permit dividing reactors intotwo categories, namely, reactors with a neutron modera­tor and breeder reactors.

A reactor designed to operate on the principle of neu­tron moderation can use natural uranium. The amountof moderator has to be such as to prohibit large numbersof neutrons from being absorbed by uranium-238 nuclei.Now there are approximately 140 times more of thesenuclei than there are uranium-235 nuclei. If there is not

F>hotons and Nuclei

~g~r Vasilievich Kurchatov (1902.1960 _ .ICISt and a remarkabl . ) promment Soviet phys-. e orgalllzer He head d th~c project in the Soviet Un" .H b e ~ wo~k of the atom-III the field of solid state ph . IOn. d e egan hIS SCIentific career

I. ySICS an set up the th f'

e ectrlcs (also known as fIt' eory 0 seIgnette-HI erroe ec TIcs) At th b . .

.. 30s he began research in the phy' f' he. egmlllng of theSICS 0 t e atomIC nucleus. Under

6. Energy Around Us 191enough moderator, the neutrons will not diminish theirspeeds to the thermal level and will be absorbed by ura­nium-238 nuclei, and the chain reaction will come to ahalt. A reactor operating on natural uranium or uraniumslightly enriched in uranium-235 will still create a newfuel, plutonium. But much less plutonium will be pro­duced than the uranium nuclei that are "burned" up.

Up until just recently, most atomic power plants usedthermal-neutron reactors. There are four types of suchreactors: water-water reactors with ordinary water as themoderator and heat-transfer agent (coolant); graphite­water reactors with water coolant and graphite moderator;reactors in which heavy water is the moderator and or­dinary water is the coolant; and, finally, graphite-gas-cooled reactors.

The reason why specialists in the field of atomic powerengineering concentrated their attention on reactors oper­ating with thermal neutrons is apparently due to thefact that it is extremely difficult to enrich uranium in the235 isotope. The reader will recall the remark madeearlier that when only the isotope uranium-235 is used,enormous reserves of potential nuclear fuel are left un-touched.

At the present time there is a tendency to have nuclearreactors operate on highly enriched fuel with no neutronmoderator.

Suppose the mixture in the reactor consists of one partof uranium-235 and one part of uranium-238. In that case,the number of neutrons that fall out of the chain reactiondue to capture by uranium-238 may exceed the number ofneutrons fissioning nuclei of uranium-235 and sustaining

his guidance, important studies were carried out in the field ofnuclear isometry, the resonance absorption of neutrons, and arti-

ficial radioactivity.

the chain reaction. Such a reactor is termed a breeder.Depending on the geometry- of the rods or bricks of nu­clear active and potential fuel, it is possible to designbreeder reactors with a great variety of percentage ratiosof the two types of fuel and with different reproductionfactors.

Two examples will suffice to give the reader some ideaof the parameters of nuclear reactors.

Figure 6.2 is a general view of the design of a nuclearreactor used in submarines of the United States. Thecoolant here is ordinary water. Since ordinary watercaptures neutrons roughly 600 times more effectively thanheavy water, such a reactor can operate only on uranium-

Photons and Nuclei

COOLANTOUTLET --\'lO'.:.;;".........

ENRICHEDURANIUM

Figure 6.2

192

CONTROL RODS

COOLANTINLET

WAll OF REACTOR

6. Energy Around Us 193

238 enriched in uranium-235. Instead of the natural por~

tion of 0.72 % in the fuel of these reactors we find fromone to four percent of uranium-235. The reactor is capableof generating 1100 megawatts of electricity, is about5 metres in diameter, has a height of 15 metres (a 5-sto­rey building) and wall thickness of about 30 cet.ttimetr.es.If such a reactor is charged with 80 tons of uranmmoxidecontaining 3.2 % of uranium-235, it will operate 10 to 12months (and then the rods have to boe replac~d). -Thewater in the reactor heats up to 320 C and circulatesunder a pressure of about 300 atmospheres. The hot waterturns to steam and is delivered to the blades of a tur­bine.

We now give a brief description of the design of apowerful breeder reactor to be built in France. It iscalled Superphenix.

The fuel will be a mixture of plutonium-239 and ura­nium-238. No moderator will be needed and so the neu­trons do not lose speed from the time of generation duringnuclear fission up to an encounter with another atomicnuclei of the fuel.

The fact that the reactor employs fast neutrons makesfor a high degree of compactness. The core of the reactorwill not exceed 10 cubic metres. Thus, there will be alarge liberation of heat per unit volume.

Heat removal cannot be done with water because watermoderates neutrons. Liquid sodium will be used. Sodiummelts at 98°C and boils at 882°C at atmosphericpressure. For technical reasons

bthe tempe.rature of liquid

sodium must not exceed 550 C. There is therefore noneed to raise the pressure of the coolant (this is resortedto when the coolant is water).

The Superphenix reactor will hav~ the following dimen­sions: inner diameter 64 metres, height about 80 metres.This is a big 20-storey buildingl The core of the reactoris in the form of a hexagonal prism made up (like a bundle

13-2542

195Photons and Nuclei 194

of pencils) of thin rods each 5.4 metres in length. Thefuel rods alternate with control rods.

There is no need (nor space) here to describe how thecore of the reactor is cooled. Suffice it to say that it isdone in three stages. A first loop contains sodium; itremoves the heat from the reactor and delivers it to theboiler, where it is taken up by a second sodium loop.In the second heat exchanger, the heat is taken up by athird loop with a circulating water-steam mixture. Thenfollows the ordinary route to the steam turbine.

Calculations show that this setup should generate3000 megawatts of thermal capacity and 1240 of electric­ity.

I cannot help stressing once again that the necessityof converting nuclear energy into electric energy passesthrough the thermal stage, leaving a poignant feeling ofdisappointment. It is something like installing an auto­mobile engine on a cart. But so far no one has come up witha better idea of how to bypass that stage, which createsperhaps· the greatest difficulties in the construction ofatomic power plants. Besides the common drawback ofall thermal power plants, we have here the added neces­sity of including intermediate pipelines. This is donebecause we have to eliminate the unacceptable radioac­tivity of the water vapour entering the turbine.

Here are some more figures of this design. The maxi­mum neutron flux per s.quare centimetre per second isequal to 6.2 X 1015 • The reproduction factor will be equalto 1.24. Used up fuel elements will be replaced oncea year. In the first loop, there will be a flow of liquidsodium (technically called the mass flow rate) of 16.4tons a second. Superheated steam will emerge under apressure of 180 bars and at a temperature of 490°C.

A few words are in order about the "ashes" of nuclearfuel. A great number of radioactive isotopes appear inthe fission of fuel nuclei. This is an uncontrollable process,

6. Energy Around Us

but we have the opportunity of obtaining any isotopeswe want merely by placing substances in the reactor.Such substances absorb neutrons and create new atoms.

Of course, radioisotopes can be obtained in accelerat?rsby subjecting various materials to bombardment WIthprotons or the nuclei of other elements. . .

The number of artificial elements obtamed to date ISvery great. The "empty" places in the periodic table ofelements have been filled up: elements No. 61, 85, and 87do not have long-lived stable isotopes, and so there arenone in nature. The Mendeleev table of elements hasbeen extended. Elements above 92 are termed transura­nium elements. Each of the transuranium elements, allthe way up to element 105, has been obtained in severalisotopic variants.

Also in addition to new chemical elements, large num­bers of'radioisotopes have been obtained of those chemicalelements that are encountered in the earth's crust intheir stable form.

Radioisotopes have been widely used for many years inthe sterilization of food products by gamma rays, in flawdetection, and as generators of electric power that useelectrons produced in decay processes. This list could beextended considerably.

The usefulness of radioactive isotopes is, unfortunately,commensurate with the trouble engineers have in pro­tecting persons that deal with radioactive radiation.

Nuclear fuel waste contains 450 kinds of atoms, includ­ing uranium-237 and neptunium-239, which convert intoneptunium-237 and plutonium-239. .

Unlike coal or petroleum, nuclear fuel does not burn upcompletely. In a number of cas.es.. nuclear reactoJs op­erate with enriched fuel contammg between 2.5 Yo and3.5 % of uranium-235. At some point, a reactor ceases todeliver energy because in the fission process a large num­ber of isotopes are produced that capture neutrons andIS·

Photons and Nuclei 197196

prevent. the fission process from developing. When areactor IS shut down, there is roughly 1% of uranium­235 and a somewhat smaller amount of plutonium-239 left.

Th~re. can be no qu~stion of throwing away such ashcontammg large quantIties of useful fuel. A big chemicalworks could be linked up with an atomic power plant.It would have to be completely automated since it wouldbe handling highly radioactive materials. Special mea­sures would also have to be taken to protect personnelfrom gamma radiation.

At the~e plants, used fuel elements would be groundup and dIssolved. The pure fuel (uranium and plutonium)could be extracted and returned for the manufacture offresh fuel elements. .

~here still. remains appreciable quantities of radio­actIve waste m the form of "hot" (radioactive) solutionsthat have to be buried. And what has to be ensured isthat for many centuries the disposal sites will not betouched in any way.

Specialists are more or less optimistic. It is believedthat ~he storing of cans (drums or barrels) of radioactivesolutIOns up to one kilometre underground in speciallyselected sp~ts shou~d guarantee one hundred percent safe­ty. What kmd of SItes are suitable? That is a matter forgeologists to decide. Only places safe from earthquakesof course. Also, one must be sure there are no subter~ranea~ rivers or streams. Salt deposits have been foundto satIsf~ such conditions. And, of course, the barrelscannot sImp.ly. be dropped into a kilometre-deep well.To .e~sure dISSIpation of heat that the barrels would beeI,tllttmg constantly, they will have to be dispersed atdIstances of at least up to 10 metres apart.

6. Energy Around Us

Thermonuclear Energy

We have already mentioned that chemical and nuclearreactions have much in common. Since heat is gen·eratednot only in reactions of decomposition but also, frequent­ly, in the combining of two molecules into one, we canexpect that atomic nuclei behave in a similar manner.

It is not difficult to see what fusion reactions (com­bining of nuclei) would be advantageous from the energypoint of view if we know the masses of the atomic nuclei.

The deuterium nucleus has a mass of 2.0146 mass units.If two nuclei fuse into one, we get 4He, which has a massof 4.0038 instead of 2 X 2.0146=4.0292. The excess massof 0.0254 is equivalent to roughly 25 MeV of energy, or4 X 10-11 joule. One gram of deuterium contains 0.3 X 1024

atoms. Now if such a reaction took place, then two gramswould yield 1013 joules of energy! It turns that the mostpromising are the fusion reactions of the heavy isotopesof hydrogen: deuterium, tritium. Even ordinary hydrogencan be put to use as thermonuclear fuel.

It is thus clear that atomic energy can be generatedin two forms: the splitting up of heavy atomic nuclei inso-called fission reactions and the combining of lightatomic nuclei in so-called fusion reactions, which are alsocalled thermonuclear reactions.

Fusion reactions, if harnessed, could ensure our powerneeds for millions of years to come (and the level of theworld's oceans would not go down perceptibly in theprocess of using the water). It would indeed be a limitlessocean of power, and at very low costs-practically freeof charge.

However, it is no easy job to harness this "free" sourceof power. The point is that all atomic nuclei are posi­tively charged. That means an enormous amount of energyis required to bring the nuclei close together.

The only way to obtain this energy is to convert a

199Photons and Nuclei 198

substance into the plasma state, which means strippingatomic nuclei of their clouds of electrons and then rais­ing the temperature of the plasma to a point where thenuclei begin to collide (collisions occur at distances of10-13 centimetre apart) despite the electric repulsiveforces.

Calculations turn out to be quite disappointing. Thereader can himself try to calculate the amount of energyof electrostatic repulsion using the formula e2 /r and thencheck the needed temperatures (you will have to recallthe formula that relates the temperature and the kineticenergy of any particle). The answer is in the tens ofmillions of degrees.

To summarize, then, we need to produce a high-tem­perature plasma. There are two ways to do this: one hasbeen in use for over two decades, the other is fifteenyears or so younger.

The former method consists in creating a thermonu­clear reactor by "driving" the plasma into a magneticbottle (that's what it's called).

If a magnetic field is imposed on a gas-discharge tubeand is made to coincide in direction with the electricfield, a plasma column is generated. We know that thecharged particles of the plasma will describe helical tra­jectories. We can say that the particles move in a singlering surface current. The stronger the magnetic field thesmaller the radius of the plasma column. The force actingon the current of charged particles via the magneticfield is what produces the column, which does not comein contact with the walls of the gas-discharge tube.

In principle, we have thus produced plasma that actu­ally "hangs in the air".

Calculations show that if the initial hydrogen pressureis of the order of 0.1 millimetre of mercury, the radiusof the column is 10 centimetres, and the intensity of thedischarge current is 500 000 amperes, then the tempera-

6. Energy Around Us

ture of the plasma should be sufficient for a fusion reac­tion to start up.

However, there are very many obstacles still in theway of a controlled thermonuclear (fusion) reaction. Theplasma column is the main stumbling block. For a num­ber of reasons it is highly unstable and disintegratesalmost instantaneously. The problem gets under controlonly if one is able to make a magnetic bottle with whatmight be called "feedback": that is, such that the randomfluctuations that disrupt the column give rise to counter­acting forces capable of restoring the column.

In the middle of 1978, a group of American physicistsworking at Princeton University succeeded in heatingplasma to 60 million degrees: This, success was a.chievedby using magnetic bottles called Tokamak (we discuss~dthem in book three of this series). Tokamak-a SovIetdevice-is an acronym that stands for toroidal, chamber,magnet. The temperature attained is sufficient for thefusion of nuclei of deuterium and tritium. Since then,Soviet and also American scientists have-attained stillhigher temperatures and better parameters for other as­pects of this extremely complicated undertaking. .

There is still a big problem, however, and that IS toconfine the hot plasma in a restricted volume for suffi­ciently long time intervals. Ways of doing this are notso obvious to the engineer. It very well may be that theachievement of controlled fusion reactions is an extreme­ly costly business. But research goes on and scientistsremain optimistic.

Attempts have recently been made to attain controlledthermonuclear fusion by means of laser emission. Lasersare now in operation with power outputs of about 1012

watts. This power can be delivered to a substance. wewish to convert to plasma in the form of pulses of lightof duration 10-9 to 10-10 second. Naturally, when light ofsuch colossal power strikes a solid, the substance is in-

201Photons and Nuclei

LITHIUM SHELL

Figure 6.3

200 6. Energy Around Us

to start up, the energy liberated in the t.ime intervalbetween the start and finish of the reactI?n must besufficient to maintain the temperature reqUIred to con­tinue the reaction. Calculations show that the plasmamust have a density from 103 to 104 tim~s that. of thesolid, or about 1026 particles to every CUbIC centJme.tre.A laser is capable of producing the .needed co~~ressIOn.

In principle, it is possible to obtam the reqmslte te.m­perature and density. Then what happens? The fUSIOnenergy of the nuclei is conveyed to the neutrons t~at.arereleased in the reaction. These neutrons fall on the lIthIUmshell of the vessel. The lithium, via a heat exchanger,transfers the energy to a turbogenerator. Part ?f. theneutrons react with the lithium and produce tntIUm,which is needed as fuel.

The principle is simple but the end is still a long wayoff, and it is quite possible that fresh and unexpectedobstacles will crop up. It is very hard to foresee ~~e re­quirements of such a reactor when power. generatIOn ac­tually begins. Research workers ar~ convmced that pro­ducing such high power outputs III small volumes ofsubstance will reveal entirely new phenomena.

stantaneously ionized and passes into the plasma state.What scientists are aiming at is a situation in which adeuterium-tritium plasma is created with a temperatureof 108 degrees, the temperature being maintained for ati~e inte~val sufficient for a chain reaction to develop.ThIs reqUIres plasma of sufficiently high density to ensurea large. enough number of collisions of the nuclei.

Those are the underlying ideas of the reactor sketchedin Figure 6.3. A solid (frozen) pellet consisting of hydro­gen isotopes falls inside a vessel that has been evacuatedto a deep vacuum. When the pellet passes through thecentre of the vessel, powerful lasers are switched on thattransform the solid into a state of plasma. For the reactor

l~

Solar Energy

The conversion of solar energy to electricity by meansof photocells has been known for a long .time, but onlyrecently has the idea occurred that thIS phenomenoncould be used as the basis for an electric power plant. ,!,hesuggestion might appear at first glance to be rather WIld.Figure it this way: to make a power plant of 1000 mega­watts, you would have to cover an area o~ 6 X 6 squarekilometres with solar cells (these are specwl photocellsfor the conversion of solar energy into electricity). Andthe calculations are for the sun-drenched Sahara desert.What would you say of a plant like that for the middle

of Europe? Here there are not so many sunny days andthe area would have to be doubled at the very least.Science fiction, pure and simple, says the reader. Justimagine what a power plant like that would cost to build!

True enough. But compare that with the advantagesof suc~ a mode of power generation. We do not use up anymatenal of the earth, we do not pollute the environmentwith any kind of waste. Aren't those two factors strongenough inducements to get down to a serious study ofways of building cheaper solar cells, optimal arrange­ments of the cells, and the focussing of solar radiation?Many scientists (and I am one of them) are convinced notonly that the problem deserves our closest study but hope

Photons and NucleI

RAY OFSUNLIGHT

METALLICELECTRODE---

METALLIC t=====::;=====::::jELECTRODE

Figure 6.4

USER

202 6. Energy Around Us 203

that precisely this path will lead to electric power plantsof the future. In a few years that might very well beproblem No. 1.

Is not this optimism rather premature? What is thesituation today in this field? First of all, what kindof solar cells does industry offer at the present time?

Figure 6.4 is a diagrammatic sketch of how solar energyis converted into electricity. The cell consists of asemiconductor p-n layer squeezed between metallic elec­trodes. The sun's rays create free electrons and holes,which are sent in opposite directions by means of a contactvoltage, thus generating a current.

There are three main types of such cells. Homocontactcells in which a p-n sandwich is created by alloyingsilicon. A diffusion process is used to make a thin (0.3 mi­crometre) n-Iayer and a relatively thick (300 micrometres)p-Iayer. Heterocontact cells consist of two different semi­conductors. An n-Iayer of cadmium sulphide is sprayedonto a metallic backing to a thickness of 20-30 micro­metres; then on this surface, chemical methods are usedto create a p-Iayer of cuprous sulphide 0.5 micrometrein thickness. A third type of cell makes use of the contactvoltage between gallium arsenide and a metal separatedby an extremely thin (20 angstroms!) dielectric film.

For optimal utilization of the energy of the wholesolar spectrum, semiconductors with electron binding en­ergies of about 1.5 electron volts are suitable. In prin­ciple, efficiencies of 28 % are attainable in solar cells.

Silicon homocontact cells, which have a number oftechnical advantages and have been studied in the mostdetail, have efficiencies from 11 % to 15 %. Silicon solarcells have been in use for more than twenty years. Thematerial here is quartz sand (silicon dioxide) from whichpure silicon is obtained. Single crystals are manufacturedwith thicknesses of about 0.3 millimetre in the shapeof circular disks. In recent years, a process has been de-

tions of solar illumination are ensured. We also have .adetailed design of a power plant to be flown on an arti­ficial earth satellite. Such space-borne power plants canreceive the energy of solar radiation with?ut a?-y lossesand then in the form of microwav.es dehver. l~ to theearth where it can be transformed mto elect~lclty. Andthis is not science fiction. Engineers are seno~sly con­sidering a satellite-borne power plant 25 by 5 kllo~et.resin size. This area could accommodate 14 000 mllhonphotocellsI The plant would weigh 100 000 tons and wouldgenerate the equivalent output of seve~al tens o~ thelargest atomic power plants now operatmg; that IS, ofthe order of ten thousand megawatts.

Detailed projects have already been worked out andsmall models are undergoing tests.

Photons and Nuclei 204

veloped for manufacturing monocrystalline tape. The tech­nology of semiconductor doping (the addition of impu­rities to a material) is well developed and permits estab­lishing a p-Iayer in the silicon disk. In order to reducethe reflection of sunlight from the silicon, the surface iscovered with a thin film of titanium oxide. With a lightintensity of 100 milliwatts per square centimetre, a disksets up a voltage of 0.6 volt. The current density of ashort circuit is equal to 34 milliamperes per square centi­metre. There are variety of ways of collecting the cellsinto batteries. Silicon monocrystalline disks are nowbeing manufactured with diameters from 5 to 7.5 centi­metres. They are fixed between plates of glass and canbe combined to form a rather powerful source of electriccurrent.

It is hoped that new technological processes will de­velop cells with a much larger area.

The chief obstacle today in the way of using solarcells for industrial power generation is the high costof manufacturing high-quality monocrystalline tape.

There is considerable hope of making solar cells outof thin polycrystalline layers. The process itself willbe cheap but the efficiency will fall drastically. Thesearch is on for cheap methods of obtaining effectivesolar cells.

At the same time, researchers are seeking ways ofincreasing the energy falling on a single cell.

There are power plant designs consisting of 34 thousand •mirrors reflecting sunlight and directing the radiationto receivers located at the top of a tower 300 metres high.

If we employ concentrated solar energy, we have to seethat the increase in the temperature of the cell does notgreatly affect the overall efficiency. In this respect, gal­lium arsenide cells have certain advantages.

Projects are now being considered of siting solar powerplants OIL the top of high mountains where good condi-

6. Energy Around Us

Figure 6.5

"-"-

205

The air masses of the earth's atmosphere are in constantmotion. Cyclones, storms, the constant trade winds of thetropics, and light breezes are only some of the greatdiversity of energy manifestations of streams of air. Theenergy of the wind has for ages been used in sailingboats and to drive windmills. The overall mean annualpower developed by all air streams over the entire globecomes to the fantastic figure of 100 000 million kilowatts.

Meteorologists are well informed about wind speedsin different places on the globe and at different alti­tudes above the earth's surface. The wind is a tempera­mental thing and so all estimates involve a mean velocityof 4 metres per second and a mean altitude of 90 metres,which is a modest figure for a coastal strip.

Apparently, the most suitable sites for utilizing the"blue" energy of the wind are coastal areas of seas andoceans. It turns out that if Great Britain took advantageof the wind even in rather calm weather (Britain is therichest in wind among the European countries), it couldgenerate roughly six times that produced by all the elec­tric power plants of the country at present. In Ireland,the energy of the winds exceeds electric consumption ofthe whole country by a factor of one hundred (true,perhaps it is not that there is so much wind but ratherso few power plants).

Some twenty or so years ago wind did not rate veryhigh as a source of future power. But the trends of powerengineering have been changing before our very eyes.Commissions are set up one after the other to look into thepossibilities of new and less expensive sources of energy.The resources of the earth are now regarded in a differentlight: humanity has begun to think about what is reason­able and what is not as far as utilization of the naturalresources hidden in the earth goes. That is why the power

of the wind is now being taken seriously. Taking a soberview of engineering possibilities, we may regard as real­istic the harnessing of a fraction of one percent of the100000 million kilowatts of available wind power. Eventhat is a good deal.

Enormous "windmills" have already been designed withvanes reaching to 100 metres and more and towers ofabout that height; windmill vane tips have been calculat­ed to reach speeds of about 500 kilometres per hour.In ordinary weather, a windmill of that class could beexpected to have a power capacity of 1 to 3 megawatts.Several thousand such windmills operating in a countrywith frequent strong winds could supply all the electric­ity needed. In Western Europe, 1261.6 X 1012 watt-hoursof electric power were generated in 1973. In principle (ifone does not skimp on initial outlay) that is but a smallpart of the energy that can be feasibly extracted fromthe wind! Gigantic windmills are alreadY under con-struction. .

Photons and Nuclei

Power from the Wind206 6. Energy Around Us

Figure 6.6

207

Photons and Nuclei 208

Calculations show that a wind-driven motor has a maxi­mum output when the rotor reduces the wind speed byone third. One should not think that wind-driven unitsmust always imitate windmills. Rotors with a verticalaxis of rotation are quite possible. The rotor shown inFigure 6.6 is capable of power outputs of the order of20 kilowatts. The advantage of such a rotor is its inde­pendence of the direction of the wind; a drawback isthat it operates only when there is a strong wind. Rotorsof this kind are manufactured with diameters of up to5.5 metres.

Quite naturally, generators driven by wind must bepositioned on a relatively small area, but still at dis­tances apart so that their interaction does not play anyrole. A power plant with a capacity of 1000 megawattsI'equires an area of about 5 to 10 square kilometres.

7. The Physicsof the Universe

Measuring Distances to the Stars

Today it is no longer possible to draw a strict boundaryline between astronomy and physics. As long as astrono­mers, like geographers, confined themselves to descrip­tions of the stellar sky, the subject of astronomy drewsmall attention from physicists. However, the _.situationchanged radically a few decades ago, particularly afterobservations began to be made from artificial earthsatellites and the moon.

Without the terrestrial atmosphere as a hindrance, itis possible to receive all signals coming to us froln thefar corners of the universe. These include fluxes of a va­riety of particles and electromagnetic radiation of practi­cally the whole spectrum, from gamma rays to radiowaves. And opportunities for observing the stars in thevisible portion of the spectrum have increased immeasur­ably.

Now the study of· particle fluxes and radiation of theelectromagnetic spectrum most definitely belong in thesphere of physics. If we also n.ote that studying. outerspace brings us face to face WIth a great dIverSIty ofphenomena that do not as yet bow to a unique interpre­tation and if we further note that we can and must beprepared for the physics of the universe. leading to thediscovery of entirely new laws of nature, It becomes clearwhy explorers of the stellar world are today mostly

14-2542

physicists-physicists by education and mode ofthought.

We begin our discussion of the universe with a classicalproblem of astronomy. Measuring distances from the earthto distant stars. At the present time, the distances be­tween the earth and the sun or the earth and the planetsare measured to a high degree of accuracy by means ofradar. The mean distance to the sun is equal to 149 573 000kilometres. It has been calculated to within a millionthfraction of the distance.

But astronomers had already measured distances insidethe solar system without the aid of radar, using a simple(in principle) method called triangulation.

7. The Physics ofthe Universe 211

Consider a group of stars, from two different spots,which should be as far apart as possible. This separationis termed the base by astronomers. As sketched in Figure7.1, astronomers first used very simple devices and, later,excellent telescopes to measure the angles between thedirections to individual stars. They noticed that it ispossible to choose a group of stars that moves across thesky as an integral whole. No matter what positions thestars are observed from, the angles between the directionsto them remain the same. But among such stars theyoften found some one star that clearly moved with respectto its neighbours. Taking one of the "fixed" stars as apoint of departure, it was possible to measure the angulardisplacement of the star that moved relative to the fixedconstellation. That angle received the name parallax.

As far back as the seventeenth century, after Galileo'sinvention of the telescope, astronomers had already mea­sured the parallaxes of the planets by observing theirdisplacements with respect to the "fixed" stars. It wasthen calculated that the earth is at a distance of 140 mil­lion kilometres from the sun. Very accurate indeed!

To the naked eye, the mutual positions of the starsremain unchanged. Only by taking photographs of thestars from different positions is it possible to detect par­allactic displacements. The largest base for such a mea­surement is the diameter of the earth's orbit. If we taketwo photographs of some portion of the sky from one andthe same observatory with a time interval of half a year,the base will be equal to nearly 300 million kilometres.

Radar cannot be used to measure stellar distances andso the diagram shown in Figure 7.1 is quite modern.

This suggests that there are stars which are in per­ceptible motion relative to other stars. But it would beextremely illogical to assume that there are fixed starsand moving stars. The conclusion we are forced to isthat the stars whose mutual positions remain unchanged

210Photons and Nuclei

Figure 7.t

14·

Photons and Nuclei 212

are much farther away than a "wandering" star (or planet).Be that as it may, with good instruments we are in aposition to measure the parallaxes of many stars.

Parallax measurements to within a hundredth of asecond of arc have been carried out for many stars. Itturned out that the stars closest to us lie at distancesgreater than one parsec.

A parsec is the distance that yields an angular dis­placement of one second if the base is the diameter of theearth's orbit. It is easy to calculate that one parsec isequal to 30 million million kilometres.

Astrophysicists also use what is known as the lightyear. The light year is the distance traversed by lightin one year. One parsec is equal to about 3.3 light years.Light travelling at 300 000 kilometres a second takeshalf a day to cross the whole solar system. We now cometo an important and amazing conclusion: our planetarysystem is very much alone in the universe.

The parallax method can be applied to distances of theorder of hundreds of light years. We then have the prob­lem of how to measure distances to more distant stars.This turns out to be not at all simple, and any confidencein the correctness of approximate estimates (for the mostpart we can only be sure of one significant digit) is ob­tained by correlating results of different measurements.

At any rate, the decisive factor in determining dis­tances to faraway stars is that there are many so-calleddouble stars among the millions upon millions of starsin our galaxy. Double stars also go by the name variable.The reason for this is clear: the brightness of the starvaries because of the mutual motions of the two stars ofthe pair. The period of variation can be measured precise­ly.

Suppose we are observing a very distant cluster ofstars. We are justified in concluding that the differencesin observed brightness are not related to any difference

7. The Physics of the Universe 213

in distance. We of course know about the inverse-squarelaw (that is, the intensity of any radiation decreaseswith the inverse square of the distance), we also knowabout brightness diminishing if light rays pass throughaccumulations of stellar gas. But if a small patch of thenight sky has been chosen and the stars are far away fromus, they can be regarded as being in identical conditions.

Studies of double stars lying in the Small MagellanicCloud led to the conclusion that there is a relationshipbetween the period of a double star and its luminosity.This means that if certain stars are at approximately thesame distance from us and the period of variation ofbrightness (this period is usually between 2 and 40 days)is the same, the stars will appear to be equally bright.

Having established this rule and being convinced of itsusefulness, we can utilize it for stars that belong todifferent clusters. By comparing stars whose periods ofvariation are the same but whose brightnesses are different,we can assert that they lie at different distances from us.In order to determine how great these distances are, wecan use the inverse-square law. But this method permitsdetermining only the relative distances of variable starsfrom the earth. We can apply it only to very distantstars and cannot use it to calibrate our information aboutthe absolute stellar-earth distances obtained in measuringparallactic displacements.

The scale for measuring distances between very distantstars was found by employing the Doppler effect.

The formulas that we discussed on page 195 of the thirdbook of this series hold true for any vibrations. Thereforethe frequencies of spectral lines observed in the spectrumof a star permit determining the velocity of the star awayfrom the earth or towards the earth. Since in the equation

Photons and Nuclei 214

c is the velocity of light (300 000 km/s), it is clear thatthe star must have a high velocity and the spectrographmust be of extremely high quality for us to detect a dis­placement of the spectral lines.

Please note that the scientist is quite certain thatthe hydrogen on the star (that is, the hydrogen that hassignalled to us) at an unimaginably colossal distance fromus is the very same hydrogen that we deal with here onthe earth. If the star were at rest, the hydrogen spectrumwould have to be exactly like the spectrum we obtainfrom a gas-discharge tube (that is how confident thephysicist is of the unity of the world!). But the lines arenoticeably shifted and stellar velocities are in the hun­dreds and even tens of thousands of kilometres a second.No one has any doubts about this explanation. Howcould there be? The spectrum of hydrogen consists of avery large number of lines, and we see the displacementof all lines (not just one) of the spectrum in perfect accordwith the Doppler equation.

But let us return to the measuring of stellar distances.Of what help can our knowledge of stellar velocities be?It is very simple. Provided of course that you have ob­served the star to have moved a certain distance duringthe year (all this of course with respect to other stars,which in this case may be regarded as "fixed"). If weknow the arc displacement cp of a star (cp being perpendic­ular to the light ray that reaches us), then, knowing thetangential velocity, we can find the distance to the starR from the formula

Rq>--=V

t

In place of t we can substitute the time of translation ofthe star.

But, says the reader, the formula involves the tangen­tial velocity, whereas we do not know the direction of

7. The Physics of the Universe 215

motion of the star. Quite true, and that is why we haveto take the following approach. A large number of doublestars with the same blinking period are selected. Thenthe radial velocities of all these stars are measured. Theradial velocity will vary from zero (if the star is movingat right angles to the ray of light) to a maximum (whenthe star is moving along the ray). If we assume that,on the average, the tangential and radial velocities arethe same, we can put the mean values of the measuredvelocities into the above formula.

The Expanding Universe

Having measured stellar distances, we can now describethe world of stars as follows. The observable portionof the universe can be broken up into an enormous numberof aggregations of stars called galaxies. Our solar systemlies in the Galaxy (our galaxy). It is called the MilkyWay and can be seen in the night sky as a large pat.ch ofgrey. The Milky Way Galaxy is in the form of a dIsk ofthickness about 100 000 light years. It contains about1011 stars of different types. The sun is one such star andlies on the outskirts of the Galaxy. The component starsare so far apart that they never collide. On the average,interstellar distances are 10 million times greater thanthe sizes of the stars themselves. To obtain a similarrarefaction in the terrestrial air space we would haveto reduce the density of the air by a factor of 1018

•

Turning now to the mutual positions of different gal­axies we find the situation is quite different. The meandista~ces between galaxies are only several times greaterthan the sizes of the galaxies themselves.

Astrophysicists have learned a great deal about thepeculiarities of mutual motions of stars belonging toa single galaxy. All this is beyond the scope of our story.

No matter, this is not the first time that we have hadto give up pictorial conceptions of the world.

Taking leave of Euclidean geometry, we can propose amodel of the universe that is simultaneously closed andyet does not have any centre or any boundaries. In sucha model, all points of space are of equ.al st.at~s. .

At first glance it would seem that Emstem IS askmgfor a great deal.' We are so used to two paral~el linesnever meeting, to the sum of the squares on the SIdes of aright triangle being equal to the square on the hypote­nuse. Yes, but let me remind you of certain lessons in geog-raphy.

On the globe of the earth, the parallels of latitudeare parallel lines. But on a map? What kind ?f ~ap?you may ask. Geographical maps are constructed m dIffer­ent ways. If we depict the earth in the form of twohemispheres, the parallels of latitude cease to be para~lellines. If we resort to what is called a rectangular proJec-

217

...--'_...

~

Figure 7.2

7. The Physics of the UniversePhotons and Nuclei 216

But in a book devoted even to the very fundamentals ofphysics we cannot help mentioning an exceptionally im­portant observation. It has been established with greatcertainty, in studies of the Doppler effect in spectrabelonging to stars of different galaxies, that all galaxiesare racing away from us in all directions. What is moreit has been shown that the velocity of recession of a galax~is directly proportional to its distance from the earth.The most distant visible galaxies are racing away fromus with velocities of recession close to half the velocityof light.

When I say "from us", it sounds strange-as if Godmade the earth and strewed stars about in the surroundingspace. That was the picture of the world in ancient times(Aristotle) and in the Middle Ages. The universe hadboundaries beyond which lay the empyrean of ancientbelief-the abode of God.

To modern man, the universe is not visualized as havingany boundaries. For if there is a boundary, what liesbeyond? We must picture the universe as being withoutbounds. On the other hand, one certainly cannot believethat the earth and sun are special bodies in the universe.This would contradict every piece of knowledge acquiredby astrophysicists. But the galaxies are racing away fromusl How can we reconcile this fact with our requirementsfor a model of the universe in which there are no bound­aries, matter is more or less uniformly distributed andthe picture of the universe as seen by an inhabita'nt ofany star is the same?

The intellectual necessity of the existence of such amodel led Einstein to the following fundamental con­clusion. Euclidean geometry that is so useful in oureveryday life does not hold true when we deal with theunimaginably great distances encountered in our studiesof the stellar world. A rejection of Euclidean geometryleads to a rejection of pictorial models of the universe.

2197. The Physics of the Universe

the same' he will arrive at the same conclusions concern­ing the ~ature of motion of the stellar world and willmeasure the same velocities of the galaxies as the earthdweller.

The model of the universe advanced by Einstein in1917 was a natural consequence of his so-called generaltheory of relativity (that part of the theory that wediscussed in Chapter 4 is called the special theory ofrelativity).

However Einstein did not envisage the possible expand-ing of a cl~sed universe. The fact that a closed universemust expand was demonstrated in 1922-1924 by the Sovietscientist Alexandr A. Friedman (1888-1925). It turnedout that the theory requires that the universe either ex­pand or have alternating e~pansions and cont~actions.At any rate it cannot be statIc. We can accept eIther oneof these two viewpoints, that is, we can assume thatwe are now living in an epoch of expansion of the universe,or we can assume that the universe was a "cosmic egg"some time ago (calculations show this time lapse to beequal to several tens of thousands .of milli.ons of years)and that it exploded and the debns has Slllce been ex-panding. .

It is important to have a clear understandlllg that ~heidea of an initial explosion is in no way connected WIththe concept of a creation of the world. It may very well bethat attempts to look too far into the future and too farinto the past and also out to excessively great distancesare unjustified within the framework of existing theories.

Let us take a simple example and examine it in thelight of a scheme that appears today to be quite reason­able. We measure the red shift of the spectral lines ofradiation coming to us from distant galaxies. Using theDoppler equation, we estimate t~e velocities of the stars.The farther they are away from us~ the faster they appearto be moving. Telescopes report speeds of recession of

Photons and Nuclei 218

tion, then the distances between latitudes cease to beequal. There is no more Euclidean geometry!

If you want to, I can show you that the Pythagoreanth~orem has failed. Take a map of important airlines(FIgure 7.2) and take the triangle Moscow-Capetown­London. On the map it turns out to be an almost exactright triangle and so the sum of the squares on the sidesmust be equal to the square on the hypotenuse. Well,y?u can guess again. Let's figure it out exactly: the'dIstance from Moscow to London is 2490 km, from Mos­cow to Capetown it is 10 130 km, and from London toCapetown 9660 km. The theorem fails completely. Ourgeometry does not hold true at all on a map. The laws ofgeometry in a plane depicting the globe differ from "or­dinary" laws.

.Take ~ look at the map of the hemispheres and you~Ill see It has "edges". But this is an illusion. Actually,If one moves over the surface of the earth, he never comesto any "edge of the earth".

There is even a joke to the effect that Einstein's sonasked his father why he was so famous. Said the greatEinstein: "I was lucky, I was the first to notice that abug crawling on the globe can crawl around the equatorand come back to the starting point." That observationdoes not represent a discovery of course. But extend itto the three-dimensional space of the universe; maintainthat the universe is finite and closed like a two-dimen­sional surface bounding the globe (thus laws of "ordinary"geometry do not hold true any longer); then from thatdraw the conclusion that all points of the universe areof an equal status in the same sense that all points onthe surface of the globe are-well, all that requires ex­ceptional intellectual courage.

Now we can conclude that if we earthlings observe thatall galaxies are racing away from us in all directionsthen an inhabitant of the planet of any star will observ~

7. The Physics ofthe Universe 221

The general theory of relativity rests on the followingprinciple: there are no experiments capable of distin­guishing the motion of bodies und~r the action of a grav­itational field from the motion of such bodies in anappropriately chosen noninertial reference frame.

Let us consider a few simple examples. We are in a liftfalling with acceleration a. Stretch out your hand anddrop abal!. How will it fall? As soon as it is released, itwill begin (according to the viewpoint of the inertialobserver) its free fall with acceleration g. Since the liftis falling with acceleration a, the acceleration with re­spect to the floor of the lift will be (g-a). An observer inthe'lift can describe the motion of the falling body withthe help of the acceleration g' =g-a. In other words,. anobserver in the lift need not speak of any acceleratIOnof the lift by "changing" the acceleration of the fieldof gravity in his frame of reference.

Now let us compare two lifts. One hangs motionless abovethe earth, the other is in motion in interplanetary spacewith acceleration a with respect to the stars. All bodiesin the lift that is stationary above the earth can fallfreely with acceleration g. And so can bodies inside theinterplanetary lift. They will "fall", with acceleration -a,to the "bottom" of the lift. Here, the bottom will be thewall opposite the direction of acceleration.

It turns out that the action of a gravitational fieldand manifestations of accelerated motion are indistin­guishable.

The behaviour of a body in a reference frame under­going accelerated motion is the same as the behaviourof a body in the presence of an equivalent gravitationalfield. However, this equivalence can be complete if weconfine ourselves to observations over small portions ofspace. Indeed, imagine a "lift" with a floor measuringthousands of kilometres. If such a lift hangs motionlessabove the earth, the phenomena taking place in it will

The General Theory of Relativity

Photons and Nuclei 220

still more distant galaxies: ten thousand kilometres asecond, a hundred thousand kilometres a second andfaster still. Now, there must be a limit to these reces~ionalspeeds. The point is that if a galaxy is moving away fromus with the velocity of light, we will in principle neverbe ab~e to see it, and all because, according to the DopplerequatIOn, the frequency of light will vanish (become zero). 'No light would ever reach us from such a galaxy.

What are the greatest distances we are capable of mea­suring with the very best of the excellent instruments atour disposal? Our estimate will of course be extremelyapproxImate. At any rate, we should not complain aboutnot being able to look far enough away; the distances weare talking about run into thousands of millions of lightyears!

To speak of still greater distances is clearly meaning­less. We might put it thus: within the framework of exist­ing ideas, to talk about distances exceeding thousands ofn;tillions of light years is physically meaningless for theSImple reason that no method has been devised for measur­ing such distances.~he situation here is much like that concerning the

traJectory of an electron: it cannot be measured for thesimple reason that the very concept is meaningless.

. The special theory of relativity made it necessary tomtroduce corrections into the laws of mechanics of bodiesmoving with velocities close to the velocity of light. Thegeneral theory oj relatiVity introduces corrections into cus­tomary concepts concerning space when we deal with O'roatdistances. It is precisely for this reason that a discu~sionof the general theory is fitting in a chapter devoted to thephysics of the universe.

be differe.nt than in t?e case where the lift is moving withacceleratIOn a relative to the fixed stars. This is clearfrom Figure 7.3: in o~e ~ase the bodies fall at an angleto the bottom of the hft, m the other case they fall verti­cally.

Thus, to summarize, the principle of equivalence holdstrue for portions of space in which the field may be re­garded as uniform.

The principle of equivalence of a gravitational fieldand a properly chosen local reference frame leads us to ani~portant conclusion: a gravitational field is linked upwIth the curvature of space and a distortion in the flowof time.. Tw? observers are engaged in measuring distances and

time m.terva!s. They are interested in events occurring ona rotatmg dIsk. One observer is located on the disk theother is motionless (relative to the stars). Incident~lly,

7. The Physics of the Universe 223

only the researcher on the disk is working. The fixedobserver merely watches the work of his colleague.

The first experiment consists in measuring the radialdistance, that is, the distance between two objects locatedon the same radius of the disk at different distances fromthe centre. The measurements are performed in the ordi­nary way: a standard ruler is laid off between the ends ofthe segment of interest. From the viewpoint of both in­vestigators, the length of the ruler laid perpendicularlyto the direction of motion is the same. And so there willbe no argument between the two observers as to thelength of the radial line-segment.

Now the disk dweller initiates his second experiment.He would like to measure the length of the circle. Theruler must now be laid down along the direction of mo­tion, and of course we have to take into account thecurvature of the circle. We therefore use a small rulerso that we can equate the length of a tangential line­segment with the length of the arc. The observers willnot argue about the number of times the ruler fitted intothe length of the circumference of the circle. And never­theless their opinions about the length of the circum­ference will differ. The point is that the fixed observerwill claim that the ruler contracted because, in the secondexperiment, it lay along the direction of motion.

And so the radius of the circle is the same for bothobservers but the length of the circumference of the circleis different. The fixed observer concludes that the formulafor the length of the circumference of a ~ircle, 21tr, isincorrect. He says that the length of the circumference if!greater than 21tr.

This example shows us how the theory of relativitycomes to a rejection of Euclidean geometry or (what isthe same thing only said in different words) to the conceptof curved space.

Similar "madness" is seen in the clock experiment.

222Photons and Nuclei

Figure 7.3

Photons and Nuclei 224

Clocks fixed at different distances .from the axis of rota­tion have different rates, and they are all slower than thefixed clock. Also, the farther a clock is from the centre ofthe disk the slower it goes. A fixed observer would say thatyou can use clocks and rulers on the disk only if you arelocated at a definite distance from the centre. Space andtime possess local peculiarities.

Now let us recall the equivalence principle. Since localpeculiarities of time and space manifest themselves ona rotating disk, this means that phenomena in a gravita­tional field behave in the same manner. The situation onthe disk is the same as in the lift depicted in Figure 7.3.Accelerated motion is indistinguishable from gravitationacting against acceleration (in the direction opposite ac­celeration).

What this means is that a local curvature of spaceand time is equivalent to the presence of a gravitationalfield.

The closed nature of the universe that we discussed inthe preceding section can undoubtedly be regarded ascorroborating the general theory of relativity.

By applying rigorous mathematical reasoning, it ispossible to derive a number of quantitative corollariesfrom the intricate equations of the general theory ofrelativity. First, Einstein demonstrated that a ray oflight passing close to the sun must be deflected. Thedeflection of a ray of light passing very close to the sunshould amount to 1.75 seconds of arc. Actual measure­ments yielded a figure of 1.70. Second, the orbit of theplanet Mercury (the perihelion of the orbit, to be precise)should turn in its plane. Calculations show that in onecentury this should amount to 43 seconds of arc. Experi­ment has revealed precisely that number. And now athird prediction that was confirmed by experiment: aphoton used up energy (and hence the frequency of thelight changes) in overcoming gravitational forces.

7. The Physics of the Universe 225

The general theory of relativity is one of the greatestattainments of human thought. It has played a tremen­dous role in the development of our ideas about the uni­verse and has revolutionized physics.

Stars of All Ages

The physics of the universe is still growing fast, itcannot at all be regarded as a fully developed sciencelike, say, the mechanics of low velocities or thermody­namics. There is therefore every reason to believe thatstellar investigations will lead to discoveries of new lawsof nature. So far this has not occurred, but the pictureof the universe drawn from time to time by the popu­larizing physicist is constantly changing. What I haveto say here and now will, in a decade or so, most likelyundergo considerable change.

Astronomers have long realized that there are differentkinds of stars. With the help of the telescope, the spec­trograph, and the interferometer we can determine manyphysical quantities that go to classify the stars.

As may be expected by analogy with terrestrial experi­ments (see page 12), the maximum intensity of the spec­trum defines the surface temperature of a star. The ob­served colour of the star is uniquely related to this tempera­ture. If the temperature is from 3 to 4 thousand degrees, thecolour is reddish, if the temperature is between 6 and 7thousand degrees, the star is yellowish. Pale blue starshave temperatures exceeding 10-12 thousand degrees.When physicists were able to get out into space, theyfound stars whose maximum radiation lies in the regionof X rays and gamma rays. This means that the tempera­tures of such stars can reach into millions of degrees.

Another important characteristic of a star is the totalenergy of the spectrum that reaches us. This is the lumi­nosity of the star. The colossal differences in luminosity15-2542

Photons and Nuclei 226

may be connected with the size of the star, its distancefrom us, and its temperature.

As to chemical composition, stars consist largely of hy­drogen-helium plasma. Our sun is a rather typical star. Itschemical composition has been determined more or lessprecisely from the type of spectrum and from theoreticalcalculations of the energy of its radiation. Hydrogenmakes up 82% and helium 18%. All other elements cometo only 0.1 % of the total mass of the sun.

The atmospheres of many stars exhibit powerful mag­netic fields that are thousands of times stronger thanthe magnetic field of the earth. We know about thisdue to spectral analysis, since the spectral lines splitin magnetic fields.

The interstellar medium is rarefied to an extent beyondall imagination: one cubic centimetre of outer space con­tains one atom. This is a good time to recall that onecubic centimetre of the air we breathe contains 2.7 X 1019

molecules, on the average. There are regions of spacewhere the density of the interstellar gas is appreciablyhigher than the average. Besides gas we encounter dust,which consists of particles of size 10-4 to 10-5 em.

It can be assumed that stars are formed out of thisgaseous-dust medium. Under the influence of gravitation­al forces, a cloud begins to contract into a. sphere. Inthe course of hundreds of thousands of years It contractsand the temperature of the star increases making it visiblein the heavens. Of course, the time required for thisdepends on the size and, hence, the mass of the con­densing cloud.

As the contraction continues, the temperature in theinterior of the star rises and attains a value at whicha thermonuclear (fusion) reaction can start up. Four nuc­lei of the hydrogen atom combine into a single nucleus ofa helium atom. Recall that in the process, 4.0339 atomicmass units of four hydrogen atoms are converted into

7. The Physics of the Universe 227

4.0038 atomic mass units of the helium atom. Thus amass equal to 0.0301 unit is converted into energy.

The burning of hydrogen that takes place in the deepinterior of a star can continue for different time intervalsdepending on the mass of the star. For the sun, this.ti~e

interval is equal to between 10 and 20 thousand millIonyears. That is the stable-state period of the star. Theforces of gravitational attraction are balanced by theinternal pressure of the hot nuclei that tend to blow upthe star. A star is hence something like a gas cylinder.The walls of the cylinder, so to speak, are the gravita­tional forces.

When the supply of hydrogen fuel comes to an end, theinternal pressure diminishes, and the core of the starbegins to collapse.

What happens next? To that question the theoreticianreplies that everything depends on whether the star is ableto shrug off its outer shell. If the star does throw off theouter shell, the mass of the star becomes roughly halfthat of the sun, and then forces take over that are capableof counteracting the gravitational forces. The result is atiny star with a high surface temperature. It goes by thename of white dwarf.

What next? Again the fate of the star depends on themass. If the white dwarf has a mass less than one and ahalf solar masses, then it will die out slowly with nodramatic events taking place. The radius will diminishand the temperature will fall. Ultimately, the dwarfwill turn into a cool star the size of the earth. Such isthe demise of most stars.

Now if the mass of the white dwarf that took shapeafter the star with fuel expended has thrown off its ol1tershell is more than one and a half solar masses, then con­traction does not come to a halt at the white-dwarf stage.Electrons merge with protons to form a neutron star,which measures only a few tens of kilometres across.15·

Photons and Nuclei 228

Calculations show that a neutron star should have atemperature of the order about ten million deoTees. Ithas a maximum radiation in the region of X ~ays.

We have just discussed the history of a star that isable to throw off the outer shell. However the mathemat­ical equations do not insist on such disrobing. Now if acelestial body retains the mass of, say, ten of our suns,then the gravitational attraction would simply destroythe star. Where the star once was there would be a blackhole.

At what stage in the compression process should thistake place, and what does the term "black hole" standfor?

Let us recall the simple laws that underlie the launchingof space craft from the earth (see the first book in thisseries). To escape from the earth's gravitational pullrequires a velocity of 11 kilometres per second. Thisvelocity is given by the equation

Mv2 ="{T

Fr~m this equation it is evident that as a sphere ofdefimte mass contracts, the escape velocity of a rocketfrom such a body into outer space will be constantlyincreasing. But the limiting velocity is equal to 300 000kilometres per second! If a stellar sphere of given masscontracts to a ball having radius

MR = "{ (300000 kmjs)2

then it becomes impossible to escape. In other words,anything at all can go into what was once a star (say,a ray of light or other electromagnetic radiation), butit cannot go out; it cannot escape from the hole. Blackhole is indeed a very appropriate designation. Using the

7. The Physics of the Universe 229

above equation, we can see at once that black holes withfrom 3 to 50 solar masses will be from 60 to 1000 kilo­metres across.

A few words are now in order about how to search forblack holes. The reader may think that this problem istoo small for a little book devoted to the whole of physics,but I think the very method of approaching such a searchis highly instructive. The talent of a scientist is revealedprecisely in the search for indirect proofs of a modelwhose properties cannot be demonstrated directly.

At first glance, the problem does indeed appear to beexceedingly complicated, if not downright unsolvable. Ablack spot in the sky 1000 kilometres across correspondsto a millionth of one second of arc. No telescope coulddetect a black hole.

Over twenty years ago the Soviet physicist Ya. Zel­dovich proposed a search for black holes based on theidea that their presence in the sky should affect the be­haviour of nearby visible bodies. Together with his co­workers, he conducted a systematic examination of starcatalogues in order to find a visible star that might berevolving about a black hole. Such a star would appearto be alone but its revolution would indicate that thespectral lines would periodically be displaced redwardsor bluewards depending on whether the star was movingaway from us or towards us.

Research workers in other countries also entered thesearch and a certain number of suitable stars were found.From the amount of the Doppler shift we can give arough estimate of the mass of the star about which thevisible satellite star is revolving. Some invisible can­didates were selected with masses three times the solarmass. Thus, these could not be white dwarfs or neutronstars.

And yet all of this is not enough to claim that suchan exotic entity as a black hole does indeed exist. Any

Radio Astronomy

231

Figure 7.4

Such telescopes are extremely sensitive..They can besteered in any direction and the antenna IS. capable ofdetecting energy fluxes of the order of 10-28 watt secondper square metre. Fantastic! .'. .

Radio astronomy has led to fundamental discovenes 1I1

the physics of the universe.

7. The Physics 01 the Universe230Photons and Nuclei

The photograph shown in Figure 7.4 depicts a parabol­ic radio antenna that focuses incident parallel radiobeams. The beams come to a point and enter a specialreceiver. The signal is then amplified by electronic equip­ment. The parabolic antenna shown in the figure is thatof the Effelsberg Radio Observatory (near Bonn, WestGermany). This fully steerable radio telescope (about100 metres in diameter) is used in joint research by teamsof scientists from many countries, including the USSR.

opponent could advance a whole series of explanationsfor the periodic Doppler shift.

However, there is one phenomenon that we can call toour aid. A black hole is capable of drawing into itself gasfrom its satellite star. As this gas falls into the blackhole, it should heat up intensely and should emit X rays.True, neutron stars and white dwarfs are also capable ofdrawing in gas, but such stars, as we have already pointedout, can be distinguished from a black hole by their mass.

Just recently a star was found that satisfied all re­quirements for the satellite of a black hole. New experi­ments will undoubtedly follow and also detailed theoret­ical calculations whose aim is to predict the peculiari­ties of the X-ray spectrum emanating from the surround­ing space of a black hole. The near future will surelyshow us how often these amazing "bodies" turn up inthe universe. There are grounds to believe that theremay be large black holes and mini-holes of mass of theorder of 10-16 gram. Such holes less than an atomic nu­cleus in size may suddenly die out and in their final actreturn the energy contained in them. This energy wouldbe so great as to satisfy all terrestrial needs for many years!What a marvelous topic for science fiction writers.

Photons and Nuclei 232

In the not too distant future radio telescopes willbe set up on the moon and on artificial earth satellites.Then the absorption and reflection of electromagneticwaves by the earth's atmosphere will cease to be an ob­stacle to the observer. So far there are two "windows"in the electromagnetic spectrum. One of them lets invisible light, the other radio waves between 2 centimetres(15000 megahertz) and 30 metres (10 megahertz).

The weather has no effect on radio-astronomy observa­tions. The radio sky is quite different from what we seeat night. Powerful radio stars, including many galaxiesand the so-called quasars (or quasi-stellar objects), arehardly at all visible in the light spectrum.

The radio emission of outer space is not very strongand its study became possible only due to the phenomenalachievements of radio electronics. Suffice it to say thatthe radio emission of the sun is a million times lesspowerful than the emission in the optical portion of thespectrum.

And yet, without radiospectroscopy we would not havebeen able to establish many important facts. For instancea ~ig ro~e in understanding processes occurring in th~umverse IS played by measurements of the residual emis­sion of explosions of supernovae.

Neutral hydrogen emits a strong wave of 21 centime­tres. Measurements of the intensity of this radio emissionhave enabled scientists to sketch a pattern of the distri­bution of interstellar gas in space and follow the move­ments of gaseous clouds.

A large number of radio galaxies and quasars have beenfound at distances oxtending to the very limit of obser­vational capabilities. Suffice it to say that the red shiftin the radiation coming from these sources reaches avalue of 3.5. The red shift is defined as the ratio of thedifference between the emitted and received wavelengthto the magnitude of the emitted wavelength. This means

,

4

7. The Physics of the Universe 233

the difference is 3.5 times greater than the wavelength ofthe radiation.

Radio methods have enabled us to look out to the veryfringes of the universe. Radio-astronomy investigationshave enabled us to clarify the nature of the cosmic ra­diation that comes to us from the outer reaches of theuniverse.

Cosmic Rays

Studies that are now conveniently conducted in outerspace beyond the earth's atmosphere demonstrate thatour planet is under a constant bombardment by streamsof nuclear particles moving at velocities close to thevelocity of light itself. These particles have energies be­tween 108 and 1020 electron volts. An energy of the orderof 1020 electron volts is eight orders of magnitude greaterthan the energy of the most powerful particle acceleratorswe have been able to build.

The primary cosmic radiation consists mainly of pro­tons (about 90%); the remainder is made up of heaviernuclei. Naturally, when cosmic-ray particles collide withmolecules, atoms, and nuclei, they can make elementaryparticles of all types. But astrophysicists are particularlyinterested in the primary radiation. How are such ener­getic particle fluxes created? Where are the sources ofcosmic rays located?

A long time ago it was demonstrated that the sun isnot the main source of cosmic radiation. But if that is so,then the responsibility for generating cosmic rays cannotbe shifted to other stars either, for they are in no waydifferent (in principle) from our sun. Then who is toblame?

In our Galaxy we have what is known as the CrabNebula that was formed in the explosion of a star in theyear 1054 (don't forget that the sky has been under ob-

Photons and Nuclei 234

servation for many thousands of years). This has beens?own to ~e ~ source of radio waves and cosmic radia­tIon-a cOlllCldence that resolves the mystery of the tre­~endous en~rgies of cosmic protons. As we know, a par­t~cle can buIld up energy by spiralling about a magnetichne of for~e. All we need to do is presume that the elec­tromagnetIC field generated in the explosion of the starplays the r~Je of a synchrotron, and then the enormousen~rgy acqUIred .by a particle as it journeys along a spiralCUI ve about a hne of force over periods of thousands ofyears can reach truly fantastic figure;;: (like those wehave mentioned).

Calcul~tions show that after covering a distance equalto t~e dIameter of our Galaxy, a cosmic particle cannotacqll:Ire more energy than 1019 electron volts. Apparently,partI~les of maximum energy come to us from othergalaXIes.. Naturally we need not assume that only stellar explo­

SIOns generate cosmic radiation. Any sources of radiowaves can also be sources of cosmic radiation.

Cosmic rays were discovered at the start of this century.~l~ctroscopes carried aloft in balloons revealed some ex­Cltlllg facts. Researchers found that the discharge of anelectroscope at high altitudes proceeds much more quick­ly than it does at sea level.

It became clear that the usual falling of the leavesof t~e electroscope is not the result of imperfections inthe lllstrument, but a phenomenon due to the action ofsome kind of external factors.. I~ t~e 1920s, physicists were quite convinced that theIOlllzatIOn of the air that removes the charge from anelectr?scope is ~~doubtedly of extraterrestrial origin. TheAmerIcan.physicist Robert Andrews Millikan (1868-1953)was the first .to state this confidently and he gave thephe~IO~enon ItS modern name: cosmic rays (or cosmicradlatlOn).

7. The Physics of the Universe 235

In 1927, the Soviet scientist D. V. Skobeltsyn obtainedthe first photograph of the tracks of cosmic rays in anionization chamber.

The energies ·of cosmic particles were determined bymethods that we have already described elsewhere. Theyturned out to be tremendous.

In their studies of cosmic rays, physicists have madea number of very remarkable discoveries. For one thing,the existence of the positron was demonstrated in pre­cisely this way. The same goes for mesons, which are parti­cles with mass intermediate between the mass of the protonand that of the electron; they too were first detectedin cosmic radiation.

Studying cosmic rays continues to be one of the mostexciting fields of physical investigation.

* * *The very incompleteness of astrophysics makes it dif­

ficult to describe it in a single chapter of a small bookdevoted to merely an introduction of the reader to thebasic facts and ideas of physics at large. I could onlychoose a few of the many physical problems that dealwith the universe, only those that I consider to be ofspecial interest.

To the Reader

Mir Publishers would be grateful for your commentson the content, translation and design of this book.We would also be pleased to receive any other sug­gestions you may wish to make.

Our address isMir Publishers2 Pervy Rizhsky Pereulok1-110, GSP. Moscow, 129820USSR

Printed in the Union of Soviet Socialist Republics

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•

OTHER BOOKS IN THE SERIES

BOOK 1 PHYSICAL BODIES

BOOK 2 MOLECULES

BOOK 3 ELECTRONS

The aim of the authors, the late Lev Landau, Nobel andLenin prize winner, and Prof. Alexander Kitaigorodsky,was to provide a clear understanding of the fundamen­tal ideas and latest discoveries in physics, set forth inreadily understandable language for the laymen. Thenew and original manner in which the material is p,;-e­sented enables this book to be enjoyed by a wide rangeof readers, from those making the first acquaintancewith physics to college graduates and others interestedin this branch of science. These books may serve as anexcellent instructional aid for physics teachers.

·1,;

l<lan NosferatuOa.u: I I

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Ex tibris]lmit •

... DtJal{ulkat{

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j,

Physical Bodies

L. D. LANDAU, A. I. KITAIGORODSKY

In this first book of the famous series Physics for Every­one by the late renowned scientist, winner of the Nobeland Lenin prizes, Academician Lev Landau, and thedistinguished physicist Alexander Kitaigorodsky, the mo­tion of bodies is dealt with from the points of view oftwo observers: one in an inertial and the other in a non­inertial frame of reference. The law of universal gravita­tion is discussed in detail, including its application forcalculating space velocities and for interpreting lunartides and various geophysical phenomena.The principal aim of the book is to explain to a laymanreader in an easy-to-understand manner the basic con­cepts and latest developments in the field of physics.